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 lily@cute.edu.tw 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. REFERENCES [1] H.-M. Lee and L. Lin, Evaluation of survey by linear order and symmetric fuzzy linguistics based on the centroid method, International Journal of Innovative Computing, Information and Control, Vol. 5, No. 12(B), pp.4945-4952, December 2009. [2] L. Lin, H.-M. Lee, Fuzzy assessment method on survey analysis, Expert Systems With Applications, Vol. 36, pp.5955-5961, April 2009. [3] L. Lin, H.-M. Lee, Fuzzy assessment for sampling survey defuzzification by signed distance method, Expert Systems With Applications, Vol. 37, pp.7852–7857, December 2010. [4] J.-S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, Vol. 116, pp. 275-288, 2000. [5] H.-J. 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