Expert Systems with Applications 36 (2009) 5955–5961

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Fuzzy assessment method on sampling survey analysis

Lily Lin a, Huey-Ming Lee b,*

a Department of International Business, China University of Technology, 56, Section 3, Hsing-Lung Road, Taipei 116, Taiwanb 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 a b s t r a c t

Keywords:Sampling surveyFuzzy logicLinguistic variable

0957-4174/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.eswa.2008.07.087

* Corresponding author. Tel.: +886 937893845; faxE-mail address: hmlee@faculty.pccu.edu.tw (H.-M

Developing a well-designed market survey questionnaire will ensure that surveyors get the informationthey need about the target market. Traditional sampling survey via questionnaire, which rates item bylinguistic variables, possesses the vague nature. It has difficulty in reflecting interviewee’s incompleteand uncertain thought. Therefore, if we can use fuzzy sense to express the degree of interviewee’s feelingsbased on his own concept, the sampling result will be closer to interviewee’s real thought. In this study,we propose the fuzzy sense on sampling survey to do aggregated assessment analysis. The proposedfuzzy assessment method on sampling survey analysis is easily to assess the sampling survey and eval-uate the aggregative evaluation.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Statistical analysis via sampling survey is a powerful market re-search tool to understand the useful information for business strat-egy plan. Traditionally, we compute statistics with sample data byasking questionnaires according to the thinking of binary logic. But,this kind of response may lead to an unreasonable bias since thehuman thinking is full with ambiguity and uncertainty. There aretwo different methods, multiple-item and single-item, while usinglinguistic variable as rating item. We use marked or unmarked todetermine the choice for each item, i.e., the marked item is repre-sented by 1, while the other unmarked item is represented by 0.Generally speaking, the linguistic variable possesses the vague nat-ure (Lin & Lee, 2008; Sun & Wu, 2006; Zadeh, 1965, 1975a, 1975b,1976). For example, the authority of one specific company decidesto conduct the satisfactory level survey on a specific article,namely, strongly dissatisfactory, dissatisfactory, middle, satisfac-tory, and strongly satisfactory. These linguistic variables are fuzzylanguages that can not be used to express the real situation by zeroor 1 to mark item. Therefore, in this paper, we apply a value mwhich belongs to the closed interval [0,1] to represent the mem-bership grade in the fuzzy sense of marked item.

In this study, we propose the fuzzy sense on sampling survey todo aggregated assessment analysis. The proposed fuzzy assess-ment method on sampling survey analysis is easily to assess thesampling survey and evaluate the aggregative evaluation. This pa-per is organized as follows. Section 1 is introduction. Section 2 isthe preliminaries, Section 3 is the sampling survey as the ‘‘multi-ple-rating choices” answer based on the crisp and fuzzy mode. Sec-

ll rights reserved.

: +886 2 2777 4723.. Lee).

tion 4 is the sampling survey as the ‘‘single” answer based on thecrisp and fuzzy case, which is the special case of ‘‘multiple-ratingchoices”. Section 5 is the determination of weights. Section 6 statesthe fuzzy aggregative assessment method. Section 7 is the numer-ical example. The conclusion of this study is given in Section 8.

2. Preliminaries

For the proposed algorithm, all pertinent definitions of fuzzysets are given below (Zadeh, 1965; Zimmermann, 1991).

Definition 2.1. If X is a collection of objects denoted generically byx then a fuzzy set eA in X is a set of ordered pairs:eA ¼ fðx;leAðxÞÞjx 2 Xg ð1Þ

leAðxÞ is called the membership function of x in eA which maps X tothe closed unit interval [0,1] that characterizes the degree of mem-bership of x in eA.

Definition 2.2. Let ea be a fuzzy set on R = (�1,1). It is called afuzzy point if its membership function is

leaðxÞ ¼ 1; if x ¼ a

0; if x–a

�ð2Þ

Definition 2.3. Let eB ¼ ða; b; cÞ, a < b < c, be a fuzzy set on R = (�1,1). It is called a normal triangular fuzzy number, if its membershipfunction is

leBðxÞ ¼x�ab�a ; if a 6 x 6 bc�xc�b ; if b 6 x 6 c

0; otherwise

8><>: ð3Þ

Table 1Contents of the proposed assessment form

Mainitem

Item -weight

Sub-item

Sub-item-weight

Linguistic variables

B1 B2 B3 � � � � � � � � � Bk

A1 a1 A11 a11

A12 a12

. .

. .

. .A1k1

a1k1

A2 a2 A21 a21

A22 a22

. .

. .

. .A2k2

a2k2

. . . .

. . . .

. . . .An an An1 an1

An2 an2

. .

. .

. .Ankn

ankn

5956 L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961

The assignment of a real value to a fuzzy number is calleddefuzzification. It can take many forms, but the most standarddefuzzification is through computing the centroid. This is defined,effectively, as the center of gravity of the curve describing a givenfuzzy quantity. Because of this definition, its computation requiresintegration of the membership functions.

Let eB ¼ ða; b; cÞ be a normal triangular fuzzy number. We let

CðeBÞ ¼ R ca xleBðxÞdxR ca leBðxÞdx

ð4Þ

be the defuzzification by the centroid of eB, where a and c are lowerand upper limits of the integral, respectively. Then, we obtain thatthe centroid of the normal triangular fuzzy number eB is

CðeBÞ ¼ R ca xleBðxÞdxR ca leBðxÞdx

¼ aþ bþ c3

ð5Þ

Definition 2.4. Fuzzy relation:

Let X,Y # R be universal sets, theneR ¼ fððx; yÞ;leRðx; yÞÞjðx; yÞ# X � Yg ð6Þ

is called a fuzzy relation on X � Y.

Proposition 2.1. Let eA1 ¼ ðp1; q1; r1Þ and eA2 ¼ ðp2; q2; r2Þ be twonormal triangular fuzzy numbers, and k > 0, then, we have

(10) eA1 � eA2 ¼ ðp1 þ p2; q1 þ q2; r1 þ r2Þ(20) k� eA1 ¼ ðkp1; kq1; kr1Þ

We can easily show the Proposition 2.1 by extension principle.

Proposition 2.2. Let eA1 ¼ ðp1; q1; r1Þ and eA2 ¼ ðp2; q2; r2Þ be twonormal triangular fuzzy numbers, and k > 0, then we have

(10) CðeA1 � eA2Þ ¼ CðeA1Þ þ CðeA2Þ(20) Cðk� eA1Þ ¼ kCðeA1Þ

We also can easily show the Proposition 2.2 by Proposition 2.1 andEq. (5).

3. The sampling survey with the multiple-rating choices answer

In most cases, questionnaire of sampling survey exists manytopics and questions, let’s say, main items and sub-items. For in-stance, one specific questionnaire regarding satisfactory levelmay include main survey items such as satisfactory level for prod-uct, service and price, etc., also sub-items may exist under eachmain item. We can define them as follows:

Main items: A1,A2, . . .,An

with weights: a1,a2, . . .,an, respectivelysubject to: 0 6 aj 6 1; j ¼ 1;2; . . . ;n, and

Pnj¼1aj ¼ 1

Table 2The sample data in crisp case, with multiple-choices from criteria B1,B2, . . .,Bk

Samples Criteria

B1 B2 � � � Bk

S1 m�ij11 m�ij12 � � � m�ij1kS2 m�ij21 m�ij22 � � � m�ij2k... ..

. ...

� � � ...

Sp m�ijp1 m�ijp2 � � � m�ijpk

Sub-items: Ai1;Ai2; . . . ;Aikiunder main items Ai; i ¼ 1;2; . . . ;n

with weights: ai1; ai2; . . . ; aiki, respectively

subject to: 0 6 aij 6 1; j ¼ 1;2; . . . ; ki, andPki

j¼1aij ¼ 1Let Bv, for v=1,2, . . .,k, be the k different linguistic variables as

criteria of questionnaire, expressed in fuzzy language such as verylow, low, medium, high, very high, etc. We describe the above asshown in Table 1.

3.1. The crisp mode

We suppose that there are n samples which are drawn from thespecific population, each sample chooses Bv, for v = 1,2, . . .,k,

respectively, the selected item denoted by 1, otherwise denotedby zero in crisp case. Sample data for each sub-item, Aij, are de-picted as shown in Table 2.

In Table 2,

m�ijuv ¼ 0 or 1; and 1 6Xk

v¼1

m�ijuv 6 k ð7Þ

for each u 2 f1;2; . . . ; pg; v 2 f1;2; . . . ; kg.

3.2. The fuzzy mode

In crisp case, for each criteria Bv only can be denoted 1, other-wise denoted zero. It has a difficulty in reflecting interviewee’sincomplete and uncertain thought. Therefore, if we can use fuzzysense of sampling to express the degree of interviewee’s feelingsbased on his own concept, the result will be closer to interviewee’sreal thought (Lin & Lee, 2008; Sun & Wu, 2006). The sample datafor each sub-item Aij in the fuzzy case are shown as Table 3.

In Table 3, mijuv 2 ½0;1�, for each u 2 f1;2; . . . ; pg; v 2 f1;2;. . . ; kg, and the following condition is satisfied,

1 6Xk

v¼1

mijuv 6 k ð8Þ

Hence, the fuzzy case in Eq. (8) corresponds to the crisp case inEq. (7).

The average of those p samples for each Bv is

mijv ¼1p

Xp

u¼1

mijuv ð9Þ

for v = 1,2, . . .,k.

Table 3The sample data in fuzzy case, each sample can select criteria Bv repeatedly bymembership grade mijuv 2 ½0; 1�

Samples Criteria

B1 B2 � � � Bk

S1 mij11 mij12 � � � mij1k

S2 mij21 mij22 � � � mij2k... ..

. ...

� � � ...

Sp mijp1 mijp2 � � � mijpk

Average mij1 mij2 � � � mijk

L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961 5957

4. The sampling survey with the single-rating choice answer

Traditional statistics deals with single-rating choice answerthrough sampling survey. The multiple-rating choices answer isthe general case of the single-rating choice answer.

4.1. The crisp mode

In the case of single choice of criteria is shown as Table 2, andEq. (7) can be re-modified as follows.

For each u 2 f1;2; . . . ; pg; v 2 f1;2; . . . ; kg, the following condi-tions are satisfied,

m�ijuv ¼ 0 or 1; andXk

v¼1

m�ijuv ¼ 1 ð10Þ

Table 5Normal triangular fuzzy numbers of grades

4.2. The fuzzy mode

For fuzzy case instead of crisp case, criteria of sample data insingle choice problem are re-modified from Table 3 as follows,and we can get Table 4.

For each u 2 {1,2, . . .,p}, v 2 {1,2, . . .,k}.Normalize mijuv in Table 3 for each sample Su, we let

Mijuv ¼mijuvPkq¼1mijuq

ð11Þ

Then, we have

Mijuv 2 ½0;1�; andXk

v¼1

Mijuv ¼ 1 ð12Þ

The average of those samples S1,S2, . . .,Sp for each Bq is

Mijv ¼1p

Xp

u¼1

Mijuv ð13Þ

for v = 1,2,. . .,k. Then, we obtain the Table 4.Hence, the fuzzy case in Eq. (12) corresponds to the crisp case in

Eq. (10).

5. Determination of weights

Experts/evaluators assign the weight, from 0 to 10, to each mainitem A1;A2; . . . ;An, or sub-item Ai1;Ai2; . . . ;Aiki

according to the rel-

Table 4The sample data in fuzzy case, with a single choice question

Samples Criteria

B1 B2 � � � Bk

S1 Mij11 Mij12 � � � Mij1k

S2 Mij21 Mij22 � � � Mij2k... ..

. ...

� � � ...

Sp Mijp1 Mijp2 � � � Mijpk

Average Mij1 Mij2 � � � Mijk

ative importance and necessity of the items. But, the crisp weight isnot so easily to be determined, for example, the weight approxi-mate 5 is more suitable than just right 5. Therefore, we use the fuz-zy numbers instead of the crisp numbers to express the weights.

We range the grade of importance of each main item or sub-item into eleven ranks, and make the linguistic values 0,1,2, . . .,10 into corresponding reasonable fuzzy numbers with normal tri-angular fuzzy numbers as listed in Table 5.

The membership functions of the normal triangular fuzzy num-bers in Table 5 are as follows:

lN0ðxÞ ¼

1� x; 0 6 x 6 10; otherwise

�

lNkðxÞ ¼

x� ðk� 1Þ; k� 1 6 x 6 k

kþ 1� x; k 6 x 6 kþ 10; otherwise

8><>:for k ¼ 1;2; . . . ;9

lN10ðxÞ ¼

x� 9; 9 6 x 6 100; otherwise

�ð14Þ

Suppose that there are r experts/evaluators assign weights Nt,t 2 {0,1,2, . . .,10} to each main item A1,A2, . . .,An, and suppose thatthe qth expert assigns weight to be the normal triangular fuzzyKqj ¼ ðfqj; gqj;hqjÞ 2 fN0;N1; . . . ;N10g for q = 1,2, . . .,r; j = 1,2, . . .,n.Then, for the item Aj, the average weight assigned by these r ex-perts/evaluators is

1r� ðK1j � K2j � � � � � KrjÞ ¼

1r

Xr

q¼1

fqj;1r

Xr

q¼1

gqj;1r

Xr

q¼1

hqj

!ð15Þ

for j = 1,2, . . .,n.Defuzzified the Eq. (15) by the centroid, we have

dj ¼13r

Xr

q¼1

ðfqj þ gqj þ hqjÞ; for j ¼ 1;2; . . . ;n ð16Þ

We let

aj ¼djPn

q¼1dq; j ¼ 1; 2; . . . ;n ð17Þ

Then,

aj 2 ½0;1� for j ¼ 1;2; . . . ;n; andXn

j¼1

aj ¼ 1 ð18Þ

and we have that the weight of the main item A1,A2, . . .,An is

a1; a2; . . . ; an ð19Þ

respectively.By the same way, suppose these r experts/evaluators assign

weight Nqi, qi 2 {0,1,2,. . .,10} to each sub-item Ai1;Ai2; . . . ;Aiki. Sup-

Grade Normal fuzzy number

0 N0 = (0,0,1)1 N1 = (0,1,2)2 N2 = (1,2,3)3 N3 = (2,3,4)4 N4 = (3,4,5)5 N5 = (4,5,6)6 N6 = (5,6,7)7 N7 = (6,7,8)8 N8 = (7,8,9)9 N9 = (8,9,10)

10 N10 = (9,10,10)

5958 L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961

pose that the qth expert assigns weights to the normal triangularfuzzy numbers Lqi1; Lqi2; . . . ; Lqiki

2 fN0;N1; . . . ;N10g, for q = 1,2,. . .,r.Let

Lqip ¼ ðfqip; gqip; hqipÞ ð20Þ

Then, the average weight of these r experts/evaluators’ assessmentfor each sub-item, Ai1;Ai2; . . . ;Aiki

is

1r� ðL1ip � L2ip � � � � � LripÞ ¼

1r

Xr

q¼1

fqip;1r

Xr

q¼1

gqip;1r

Xr

q¼1

hqip

!ð21Þ

for p = 1,2, . . .,ki.Defuzzified Eq. (21) by the centroid, we have

eip ¼13r

Xr

q¼1

ðfqip þ gqip þ hqipÞ; for p ¼ 1;2; . . . ; ki ð22Þ

Let

aip ¼eipPkiu¼1eiu

2 ½0;1�; for p ¼ 1;2; . . . ; ki ð23Þ

Then, for each i 2 f1;2; . . . ; ng;Pki

p¼1aip ¼ 1Therefore, we have that the weight of the sub-item

Ai1;Ai2; . . . ;Aikiis

ai1; ai2; . . . ; aiki; ð24Þ

respectively.

6. The fuzzy aggregative assessment for sampling survey

Let B = {B1,B2, . . .,Bk} be the set of the criteria rating for eachsub-item.

6.1. The single-rating choice mode in fuzzy case

Applying with the rules of fuzzy aggregative assessment basedon Table 4, we have the following analysis results.

For each i 2 {1, 2, . . . , n}, by the fuzzy relation on Aij � B, and Eq.(13), we can form a fuzzy assessment row vector Rij as follows:

Rij ¼ ðMij1;Mij2; . . . MijkÞ ð25Þ

for j = 1,2, . . .,ki.From Eq. (25), and by fuzzy relation on Ai � B, we can form a

fuzzy assessment matrix Ri as follows:

ð26Þ

for i = 1,2, . . .,n.

(1) Evaluating the first-stage aggregative assessment for mainitem Ai, we have

ðai1;ai2; . . . ;aikiÞ Ri ¼ ðai1;ai2; . . . ;aiki

Þ

Mi11 Mi12 � � � Mi1k

Mi21 Mi22 . . . Mi2k

..

.

Miki1 Miki2 . . . Mikik

0BBBB@1CCCCA

¼ ðbi1;bi2; . . . ;bikÞ ¼bi1

B1þ bi2

B2þ � � � þ bik

Bkð27Þ

where

biq ¼minXki

u¼1

aiuMiuq;1

( ); for q ¼ 1;2; . . . ; k ð28Þ

i.e., for the ith main item Ai, the membership grade of the criterionBq is biq.We denote RðiÞ1 ¼ ðbi1; bi2; . . . ; bikÞ the vector of the first-stage aggre-gative assessment for the main item Ai with respective to the set ofthe criteria rating {B1,B2, . . .,Bk}.We let

Piq ¼biqPku¼1biu

2 ½0;1�; for q ¼ 1;2; . . . ; k ð29Þ

We can have that the aggregated assessment of presumptive ratefor the criterion Bq of the ith main item Ai is as follows:

Piq � 100%; for q ¼ 1;2; . . . ; k ð30Þ

(2) Let A = {A1,A2, . . .,An}. From Eq. (27), and by the second-stageassessment method, we can form a fuzzy assessment matrixR as follows. Let

ð31Þ

We can compute the fuzzy aggregative assessment for all of sam-ples as follows:

ða1; a2; . . . ; anÞ R ¼ ða1; a2; . . . ; anÞ

b11 b12 . . . b1k

b21 b22 . . . b2k

..

.

bn1 bn2 . . . bnk

0BBBB@1CCCCA

¼ ðb1; b2; . . . ; bkÞ ¼b1

B1þ b2

B2þ � � � þ bk

Bkð32Þ

i.e., for the criterion Bq, the membership grade is bq, where

bq ¼minfXn

u¼1

aubuq;1g; for q ¼ 1;2; . . . ; k ð33Þ

We denote R2 = (b1,b2, . . .,bk) the vector of the second-stage aggre-gative assessment with respective to the set of the criteria rating{B1,B2, . . .,Bk}.Let

Pq ¼bqPku¼1bu

; for q ¼ 1;2; . . . ; k ð34Þ

i.e., for the criterion Bq, the aggregated assessment of the presump-tive rate is

Pq � 100%; for q ¼ 1;2; . . . ; k ð35Þ

Then, we have the following proposition.

Proposition 6.1. For the sampling survey with main items A1,A2

, . . .,An which are with weights a1,a2, . . ., an, respectively, and for eachitem Ai, there are some sub-items, saying, Ai1;Ai2; � � �Aiki

with weightsai1; ai2; � � � aiki

respectively. And, there are some criteria with linguisticvariables, saying B = {B1,B2, . . .,Bk}, for the single-rating answer in thefuzzy cases, we have the following results:

(1) For the sub-item Aij, the row vector the fuzzy relation Rij onAij � B is as shown in Eq. (25).

(2) For the main item Ai, the fuzzy aggregative assessment is shownin Eq. (27).

(3) For the main item Ai, the presumptive rate for the criterionBq isPiq � 100%, for q=1,2, . . ., k, where Piq is shown in Eq. (29).

L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961 5959

(4) For the aggregative presumptive rate for the criterion Bq isPq � 100%; q ¼ 1;2; . . . ; k, where Pq is shown in Eq. (34).

6.2. The multiple-rating choices mode in fuzzy case

Based on Table 3, we can have the following analysis result.For each i 2 {1,2, . . .,n}, by fuzzy relation on Aij � B, and Eq. (9),

we can form a row vector R�ij of the fuzzy assessment as follows:

ð36Þ

for j = 1,2, . . .,k.For the sub-items Ai1;Ai2; . . . ;Aiki

of the main item Ai, from Eq.(36) and by the fuzzy relation on Ai � B, we can form a fuzzyassessment matrix R�i as the following:

ð37Þ

(1) Evaluating the first-stage aggregative assessment for mainitem Ai, we have

ðai1;ai2; . . . ;aikiÞ R�i ¼ ðai1;ai2; . . . ;aiki

Þ

mi11 mi12 � � � mi1k

mi21 mi22 . . . mi2k

..

.

miki1 Miki2 . . . Mikik

0BBBB@1CCCCA

¼ ðb�i1;b�i2; . . . ;b�ikÞ ¼

b�i1B1þ b�i2

B2þ � � � þ b�ik

Bkð38Þ

where

b�iq ¼minXki

u¼1

aiumiuq;1

( ); for q ¼ 1;2; . . . ; k ð39Þ

We denote R�ðiÞ1 ¼ ðb�i1; b�i2; . . . ; b�ikÞ the vector of the first-stage aggre-

gative assessment for the main item Ai with respective to the set ofthe criteria rating {B1,B2, . . .,Bk}.Let

P�iq ¼b�iqPku¼1b�iu

2 ½0;1� ð40Þ

for q = 1,2, . . .,k, i.e., for the main item Ai, the presuming rate for thecriterion Bq is P�iq � 100%, for q = 1,2, . . .,k.

(2) From Eq. (38), and by the fuzzy relation on A � B and by thesecond-stage assessment method, we can form a fuzzyassessment matrix R* as follows.Let

ð41Þ

We can compute the fuzzy aggregative assessment for all of sam-ples as follows:

ða1; a2; . . . ; anÞ R� ¼ ða1; a2; . . . ; anÞ

b�11 b�12 . . . b�ikb�21 b�22 . . . b�2k

..

.

b�n1 b�n2 . . . b�nk

0BBBB@1CCCCA

¼ ðb�1; b�2; . . . ; b�kÞ ¼

b�1B1þ b�2

B2þ � � � þ b�k

Bkð42Þ

where

b�q ¼Xn

u¼1

aub�uq; q ¼ 1;2; . . . ; k ð43Þ

We denote R�2 ¼ ðb�1; b

�2; . . . ; b�kÞ the vector of the second-stage aggre-

gative assessment with respective to the set of the criteria rating{B1, B2, . . ., Bk}.Let

P�q ¼b�qPKu¼1b�u

2 ½0;1� ð44Þ

for q = 1, 2, . . ., k, i.e., the aggregated assessment presuming rate forthe criterion Bq is

P�q � 100%; for q ¼ 1;2; . . . ; k ð45ÞThen, we have the following proposition.

Proposition 6.2. For the sampling survey with main itemsA1,A2, . . .,An with weights a1,a2, . . ., an, respectively, and for each itemAi, there are some sub-items, saying Ai1;Ai2; � � �Aiki

with weightsai1; ai2; � � � aiki

, respectively, and there are some criteria with linguisticvariables, saying B = {B1,B2, . . .,Bk}. For the multiple-rating answer inthe fuzzy cases, we have the following results:

(1) For the sub-item Aij, the row vector R�ij of the fuzzy relation onAij � B is as shown in Eq. (36).

(2) For the main item Ai, the fuzzy Aggregative assessment is shownin Eq. (38).

(3) For the main item Ai, the presuming rate for the criterion Bq isP�iq � 100%; q ¼ 1;2; . . . ; k, where P�iq as shown in Eq. (40).

(4) For the aggregative presuming rate for the criterion Bq isP�q � 100%; q ¼ 1;2; . . . ; k, where P�q is as shown in Eq. (44).

7. Numerical example

In this section, we use example (Lin & Lee, 2008) as shown inFig. 1 to illustrate the rate of aggregative investment benefit eval-uation process under the fuzzy environment.

The criteria ratings of benefit are linguistic variables with lin-guistic values B1,B2, . . .,B7, where B1 = extra low, B2 = very low,B3 = low, B4 = middle, B5 = high, B6 = very high, B7 = extra high.The triangular fuzzy number representations of the linguistic val-ues are shown in Table 6.

Example. Assume that there are two experts/evaluators, saying E1

and E2, and each of them assesses attributes, weights, grade ofinvestment benefit for each investment benefit factor item, asshown in Tables 7 and 8, respectively.

Then, by Eq. (8), we have the average of these two evaluators’data for each item as shown in Table 9.

Based on the proposed algorithm in Section 6, we have

(1) The row vectors of fuzzy relation on Aij � B (for i = 1,2;k = 1,2) are as follows:

R11 ¼ ð0:085;0:815;0:1;0; 0;0; 0ÞR12 ¼ ð0:075;0:825;0:1;0; 0;0; 0ÞR21 ¼ ð0;0:65; 0:35; 0;0; 0;0ÞR22 ¼ ð0:1;0:845;0:055;0; 0;0; 0ÞR31 ¼ ð0:125;0:875;0;0; 0;0; 0ÞR32 ¼ ð0;0:725;0:225;0:05;0; 0;0ÞR41 ¼ ð0:175;0:475;0:35;0; 0;0;0ÞR42 ¼ ð0:1;0:8; 0:1; 0;0; 0;0ÞR51 ¼ ð0;0:675;0:25;0:075;0; 0;0ÞR52 ¼ ð0;0:6;0:35; 0:05;0; 0;0Þ

X5: Politics

X1: L a bor

X51: Regulatory restrictions level

X52: Investment subsidy level

X2: Geography

X4: Reward

X3: Economic

X11

: Salary level

X42: Institution perform level

X41: Reward obtain level

X32: The index of industry modern times

X31: The index of industry production growth

X22: Nearing market level of delivery system

X21: Usage condition level of factory place

X12: Manpower level

Aggregative benefit

Fig. 1. Hierarchical structure model of aggregative benefit (Lin & Lee, 2008).

Table 6Normal fuzzy numbers of the criteria of benefit

Rating of benefit Fuzzy number

B1 D1 = (0.0,0.0,1/6)B2 D2 = ( 0.0,1/6,2/6)B3 D3 = ( 1/6,2/6,3/6)B4 D4 = (2/6,3/6,4/6)B5 D5 = (3/6,4/6,5/6)B6 D6 = (4/6,5/6,1.0 )B7 D7 = (5/6,1.0,1.0)

5960 L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961

(2) The fuzzy aggregative assessment for the main items are asfollows:

Table 7Content

Attribu

X1

X2

X3

X4

X5

(a) the aggregative assessment for main item A1 isRð1Þ1 ¼ ð0:08033;0:81967;0:1;0;0;0;0Þ,

(b) the aggregative assessment for main item A2 isRð2Þ1 ¼ ð0:0474;0:74243;0:21017;0;0;0;0Þ,

(c) the aggregative assessment for main item A3 isRð3Þ1 ¼ ð0:04375;0:7775;0:14625;0:0325;0; 0;0Þ,

s of the assessment form for evaluator E1

te Benefit item Weight 2 Weight 1 Linguistic va

B1

(0,1,2)X11 (4,5,6) 0.17X12 (2,3,4) 0

(2,3,4)X21 (1,2,3) 0X22 (4,5,6) 0

(3,4,5)X31 (2,3,4) 0.15X32 (6,7,8) 0

(3,4,5)X41 (5,6,7) 0.25X42 (7,8,9) 0

(5,6,7)X51 (3,4,5) 0X52 (1,2,3) 0

(d) the aggregative assessment for main item A4 isRð4Þ1 ¼ ð0:13915;0:63035;0:2305;0;0;0;0Þ,

(e) the aggregative assessment for main item A5 isRð5Þ1 ¼ ð0;0:6515625;0:281254;0:0671875;0;0;0Þ.

(3) For the main item A1, the presuming rate for the criterion B1

is 8.0333%, B2 is 81.967%, B3 is 10%; B4, B5, B6, and B7 are 0%.(4) For the aggregative presuming rate for the criterion

(B1,B2,B3,B4,B5,B6,B7), we have that it corresponds to(0.054877,0.712225,0.206069,0.026829,0,0,0).

From the above (4), we can represent the aggregative invest-ment benefit as the follows:

eI ¼ 0:054877B1

þ 0:712225B2

þ 0:206069B3

þ 0:026829B4

þ 0B5þ 0

B6þ 0

B7

Defuzzified the above formula, we have that the rate of aggregativeinvestment benefit is

CðeIÞ ¼ 0:2038766

riables

B2 B3 B4 B5 B6 B7

0.83 0 0 0 0 00.8 0.2 0 0 0 0

0.6 0.4 0 0 0 00.89 0.11 0 0 0 0

0.85 0 0 0 0 00.75 0.25 0 0 0 0

0.75 0 0 0 0 00.8 0.2 0

0.75 0.25 0 0 0 00.6 0.4 0 0 0 0

Table 8Contents of the assessment form for evaluator E2

Attribute Benefit item Weight 2 Weight 1 Linguistic variables

B1 B2 B3 B4 B5 B6 B7

X1 (2,3,4)X11 (2,3,4) 0 0.8 0.2 0 0 0 0X12 (3,4,5) 0.15 0.85 0 0 0 0 0

X2 (1,2,3)X21 (7,8,9) 0 0.7 0.3 0 0 0 0X22 (3,4,5) 0.2 0.8 0 0 0 0 0

X3 (2,3,4)X31 (3,4,5) 0.1 0.9 0 0 0 0 0X32 (5,6,7) 0 0.7 0.2 0.1 0 0 0

X4 (1,2,3)X41 (5,6,7) 0.1 0.2 0.7 0 0 0 0X42 (2,3,4) 0.2 0.8 0 0

X5 (2,3,4)X51 (6,7,8) 0 0.6 0.25 0.15 0 0 0X52 (2,3,4) 0 0.6 0.3 0.1 0 0 0

Table 9Contents of the aggregative assessment form

Attribute Benefit item Weight 2 Weight 1 Linguistic variables

B1 B2 B3 B4 B5 B6 B7

X1 (1,2,3)X11 (3,4,5) 0.085 0.815 0.1 0 0 0 0X12 (2.5,3.5,4.5) 0.075 0.825 0.1 0 0 0 0

X2 (1.5,2.5,3.5)X21 (4,5,6) 0 0.65 0.35 0 0 0 0X22 (3.5,4.5,5.5) 0.1 0.845 0.055 0 0 0 0

X3 (2.5,3.5,4.5)X31 (2.5,3.5,4.5) 0.125 0.875 0 0 0 0 0X32 (5.5,6.5,7.5) 0 0.725 0.225 0.05 0 0 0

X4 (2,3,4)X41 (5,6,7) 0.175 0.475 0.35 0 0 0 0X42 (4.5,5.5,6.5) 0.1 0.8 0.1 0

X5 (3.5,4.5,5.5)X51 (4.5,5.5,6.5) 0 0.675 0.25 0.075 0 0 0X52 (1.5,2.5,3.5) 0 0.6 0.35 0.05 0 0 0

L. Lin, H.-M. Lee / Expert Systems with Applications 36 (2009) 5955–5961 5961

8. Conclusion

Based on the above assessment method, we may apply thefuzzy relation to derive the aggregative evaluation, such as therate of aggregative risk (Lee, 1996), the facility site selection(Lin & Lee, 2008), etc. An alternative to multiple-choice testingis suggested for facility site selection assessment. The proposedfuzzy assessment method on sampling survey analysis is easilyto assess the sampling survey and evaluate the aggregativeevaluation.

Acknowledgement

The author would like to express his gratitude to Professor Jing-Shing Yao for his helpful suggestions.

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