Fuzzy interpolative reasoning based on the ratio of fuzziness of rough-fuzzy sets

Information Sciences 299 (2015) 394–411

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Fuzzy interpolative reasoning based on the ratio of fuzziness of rough-fuzzy sets Shyi-Ming Chen a,⇑, Shou-Hsiung Cheng b,c, Ze-Jin Chen a a

Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Information Management, Chienkuo Technology University, Changhua, Taiwan c Department of Kinesiology Health Leisure Studies, Chienkuo Technology University, Changhua, Taiwan b

a r t i c l e

i n f o

Article history: Received 9 August 2014 Received in revised form 27 October 2014 Accepted 2 December 2014 Available online 12 December 2014 Keywords: Degrees of fuzziness Fuzzy interpolative reasoning Rough-fuzzy sets Piecewise fuzzy entropy Ratio of fuzziness

a b s t r a c t In this paper, we propose a method to construct a polygonal rough-fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts’ opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets, where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough-fuzzy sets. The proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough-fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Fuzzy interpolative reasoning is a very important research topic in sparse fuzzy rule-based systems. In recent years, some fuzzy interpolative reasoning methods have been presented for sparse fuzzy rule-based systems, summarized as follows: (1) The fuzzy interpolative reasoning methods presented in [3,4,10,11,15–17,20,23–25,31,34–38,40–45,48,49,55–66] are based on type-1 fuzzy sets [68]. (2) The fuzzy interpolative reasoning methods presented in [14,18,19,21,26,27] are based on interval type-2 fuzzy sets [51]. Type-2 fuzzy sets and interval type-2 fuzzy sets have been used in many applications [5–9,27– 30,32,33,39,46,47,50,52,53,67,69]. (3) The fuzzy interpolative reasoning method presented in [13] is based on rough-fuzzy sets [1]. Table 1 shows a comparison of the characteristics of the existing fuzzy interpolative reasoning methods. In the following, we analyze the existing fuzzy interpolative reasoning methods as follows: (1) The fuzzy interpolative reasoning methods presented in [2,10,23,37,38,59,60] deal with fuzzy rules interpolation with complicated membership functions, such as polygonal membership functions or bell-shaped membership functions, and can deal with fuzzy rules interpolation with multiple antecedent variables. ⇑ Corresponding author. Tel.: +886 2 27376417; fax: +886 2 27301081. E-mail address: [email protected] (S.-M. Chen). http://dx.doi.org/10.1016/j.ins.2014.12.005 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

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(2) The fuzzy interpolative reasoning methods presented in [2–4,23,31,37,43,48,55,57–60] deal with fuzzy rules interpolation based on the two-fuzzy-rules interpolation scheme. The fuzzy interpolative reasoning methods presented in [1,10,16,17,25,38,42,44,63–65] deal with fuzzy rules interpolation based on the multiple-fuzzy-rules interpolation scheme. (3) The fuzzy interpolative reasoning methods presented in [16,17,49] allow the variables appearing in the antecedents of fuzzy rules to have different weights for weighted fuzzy interpolative reasoning. (4) The fuzzy interpolative reasoning methods presented in [18,19,21,26] deal with fuzzy rules interpolation in sparse fuzzy rule-based systems based on interval type-2 fuzzy sets. (5) The fuzzy interpolative reasoning methods presented in [40] deal with the backward fuzzy interpolation in sparse fuzzy rule-based systems. (6) The fuzzy interpolative reasoning methods presented in [61–65] deal with the adaptive fuzzy interpolation in sparse fuzzy rule-based systems. (7) The fuzzy interpolative reasoning method presented in [66] not only can be represented in a closed form but also guarantees that the interpolated results are valid fuzzy sets. However, the linguistic values of the linguistic variables appearing in the fuzzy rules of the existing fuzzy interpolative reasoning methods [3,4,10,11,14,15,17,21,23,22–27,31,34–38,40–45,48,49,55–66] are given by a single expert represented by type-1 fuzzy sets [68] or interval type-2 fuzzy sets [51]. If the linguistic values of the linguistic variables appearing in the fuzzy rules can be the aggregation of multiple experts’ opinions represented by rough-fuzzy sets, then there is room for more flexibility. Therefore, in [13], Chen and Shen presented a fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on rough-fuzzy sets [1], which allows the linguistic values of the linguistic variables appearing in the fuzzy rules can be the aggregation of multiple experts’ opinions. In [13], Chen and Shen pointed out that the representation of a concept should satisfy the requirements of not only the imprecise description, but also both of the common perception and the individual perception. For example, Fig. 1 shows different membership functions with respect to a specific concept defined by different people, where F1, F2, . . ., Fh1 and Fh are triangular fuzzy sets. From Fig. 1, we can see that it is difficult to describe this specific concept using conventional fuzzy sets. The definition of the rough-fuzzy sets [13] can be used to represent this uncertain concept. Based on the definition of rough-fuzzy sets, two approximations of a rough-fuzzy set can be constructed, as shown in Fig. 2 [13], where the lower approximation indicates the common region that is agreed by each person and the upper approximation indicates the individual region that is given by at least one person. In [13], Chen and Shen pointed out that rough-fuzzy sets use the lower approximation and the upper approximation to express different types of uncertainty involved in defining fuzzy memberships. In [13], Chen and Shen presented a fuzzy interpolative reasoning method based on triangular rough-fuzzy sets. However, Chen and Shen’s fuzzy interpolative reasoning method [13] has the following two drawbacks: (1) Chen and Shen’s method [13] only can deal with fuzzy interpolative reasoning using triangular rough-fuzzy sets. (2) Chen and Shen’s method [13] gets unreasonable fuzzy interpolative reasoning results in some situations. Therefore, we need to develop a new fuzzy interpolative reasoning method based on polygonal rough-fuzzy sets to overcome the drawbacks of Chen and Shen’s method [13]. In this paper, we propose a method to construct a polygonal rough-fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts’ opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets, where the values of the antecedent variables and the values of the consequence variable appearing in the fuzzy rules are represented by the cone ij and the structed polygonal rough-fuzzy sets. First, the proposed method constructs the upper approximation fuzzy set A e ij of each antecedent polygonal rough-fuzzy set A e ij , constructs the upper approximation lower approximation fuzzy set A e  and the lower approximation fuzzy set A e  of each observation polygonal rough-fuzzy set A e  , and constructs fuzzy set A j j j e i and the lower approximation fuzzy set B e i of the consequence polygonal rough-fuzzy the upper approximation fuzzy set B e set B i of each fuzzy rule Rule i, where 1 6 i 6 q and 1 6 j 6 m. Then, it calculates the weights of the fuzzy rules, respectively.

Table 1 A comparison of the characteristics of the existing fuzzy interpolative reasoning methods. Methods

Characteristics

Fuzzy interpolative reasoning methods presented in [3,4,10,11,15–17,20,23–25,31,34–38,40–45,48,49,55–66] Fuzzy interpolative reasoning methods presented in [14,18,19,21,26,27] Fuzzy interpolative reasoning method presented in [13]

The values of the linguistic variables appearing in the fuzzy rules are decided by a single expert and are represented by type-1 fuzzy sets [68] The values of the linguistic variables appearing in the fuzzy rules are decided by a single expert and represented by interval type-2 fuzzy sets [51] The values of the linguistic variables appearing in the fuzzy rules are decided by multiple experts and represented by rough-fuzzy sets [1]

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1

F1 F2

...

0

Fh

X

Fig. 1. Different membership functions with respect to a specific concept defined by different people.

1

0

X

Fig. 2. The upper approximation and the lower approximation of a rough-fuzzy set obtained from the membership functions shown in Fig. 1 [13].

e i of each Then, based on the obtained weights of the fuzzy rules and the obtained consequence polygonal rough-fuzzy set B 0 e fuzzy rule Rule i, it gets the intermediate polygonal rough-fuzzy set B . Then, it calculates the degree of fuzziness of the antee ij of each fuzzy rule Rule i and calculates the degree of fuzziness of the observation polygcedent polygonal rough-fuzzy set A e  , respectively, where 1 6 i 6 q and 1 6 j 6 m. Then, it calculates the average degree of fuzziness of the onal rough-fuzzy set A j

e ; A e ; . . . ; A e e observation polygonal rough-fuzzy sets A 1 2 m1 and A m and calculates the average degree of fuzziness of the antee e e e cedent polygonal rough-fuzzy sets A i1 ; A i2 ; . . . ; A iðm1Þ and A im of each fuzzy rule Rule i, respectively, where 1 6 i 6 q and 1 6 j 6 m. Then, it calculates the ratio of the average degree of fuzziness of the observation polygonal rough-fuzzy sets to the average degree of fuzziness of the antecedent polygonal rough-fuzzy sets of the fuzzy rules. Finally, based on the obtained ratio of the average degree of fuzziness of the observation polygonal rough-fuzzy sets to the average degree of fuzziness of the antecedent polygonal rough-fuzzy sets of the fuzzy rules and the obtained intermediate polygonal rough-fuzzy e 0 , it gets the fuzzy interpolative result B e  represented by a polygonal rough-fuzzy set. set B The main contribution of this paper is that we propose a method to construct a polygonal rough-fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts’ opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets, where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough-fuzzy sets. The proposed fuzzy interpolative reasoning method can overcome the drawbacks of Chen and Shen’s method [13] due to the fact that the proposed method deals with fuzzy interpolative reasoning using polygonal rough-fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than Chen and Shen’s method [13]. The rest of this paper is organized as follows. In Section 2, we briefly review Chen and Shen’s method [13] for constructing a triangular rough-fuzzy set from a set of triangular fuzzy sets. Moreover, we also propose a method for constructing a polygonal rough-fuzzy set from a set of polygonal fuzzy sets. In Section 3, we propose the definition of the representative value of a polygonal rough-fuzzy set and propose the definition of the degree of fuzziness of a polygonal rough-fuzzy set. In Section 4, we propose a new fuzzy interpolative reasoning method based on the ratio of fuzziness of the constructed polygonal roughfuzzy sets. In Section 5, we use some examples adopted from [13] to compare the experimental results of the proposed fuzzy interpolative reasoning method with the ones of Chen and Shen’s method [13]. The conclusions are discussed in Section 6. 2. Preliminaries In this section, we briefly review Chen and Shen’s method [13] for constructing a triangular rough-fuzzy set from a set of triangular fuzzy sets. Moreover, we also propose a method for constructing a polygonal rough-fuzzy set from a set of polygonal fuzzy sets. 2.1. Constructing a triangular rough-fuzzy set from a set of triangular fuzzy sets In [13], Chen and Shen presented a method to construct a triangular rough-fuzzy set from a set of triangular fuzzy sets. Fig. 3(a) shows three triangular fuzzy sets F1, F2 and F3 defined by three people with respect to the fuzzy concept ‘‘warm’’,

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F; e F > from the triangular fuzzy sets F1, F2 and F3 [13]. Fig. 3. Constructing a triangular rough-fuzzy set e F ¼< e

respectively. In order to capture the meaning of the fuzzy concept ‘‘warm’’, they presented a method to construct a triangular rough-fuzzy set e F ¼< e F; e F > from the triangular fuzzy sets F1, F2 and F3 shown in Fig. 3(a). First, they get the lower approximation e F of the triangular rough-fuzzy set e F ¼< e F; e F >, where e F is obtained from the intersection of the triangular fuzzy F ¼ ða0 ; a1 ; a2 Þ of the triansets F1, F2 and F3 shown in Fig. 3(a). Then, they get the top point a1 of the upper approximation e P3 e e e gular rough-fuzzy set F ¼< F ; F >, where a1 ¼ c¼1 wc ac1 , ac1 represents the top point of the cth triangular fuzzy set Fc = (ac0, F ¼< e F; e F > is ac1, ac2), wc is the weight assigned to ac1 ; wc ¼ 13, and 1 6 c 6 3. Finally, the resulting triangular rough-fuzzy set e shown in Fig. 3(b). It should be noted that triangular rough-fuzzy sets are special cases of polygonal rough-fuzzy sets. In the next subsection, we will propose a method to construct a polygonal rough-fuzzy set from a set of polygonal fuzzy sets.

2.2. Constructing a polygonal rough-fuzzy set from a set of polygonal fuzzy sets In this section, we propose a method for constructing a polygonal rough-fuzzy set from a set of polygonal fuzzy sets. Fig. 4 shows different membership functions of polygonal fuzzy sets with respect to a specific concept defined by different people, where A1, A2, . . ., Ah1 and Ah are convex polygonal fuzzy sets. Let Ak = (ak,0, ak,1, . . ., ak, l, ak, r, . . ., ak, n2, ak, n1; lk,1, . . ., lk, l, lk, r, . . ., lk, n2, lk, n1) be the kth polygonal fuzzy set of the polygonal fuzzy sets A1, A2, . . ., Ah1 and Ah, where 1 6 k 6 h, the degrees of membership of the characteristic points ak,0, ak,1, . . ., ak, l, ak, r, . . ., ak, n2 and ak, n1 are lk,0, lk,1, . . ., lk, l, lk, r, . . ., lk, n1 n1 n2 and lk, n1, respectively, lk;0 ¼ lk;n1 ¼ 0; lk;l ¼ lk;r ; 0 6 lk;l 6 1; 0 6 lk;r 6 1; l ¼ b 2 c; r ¼ d 2 e, and n P 1. In Fig. 4, ak,0 and ak, n1 are called the ‘‘left extreme point’’ and the ‘‘right extreme point’’, respectively, ak, l and ak, r are called the ‘‘left c, and r ¼ dn1 e. normal point’’ and the ‘‘right normal point’’, respectively, where n P 1; 1 6 k 6 h; l ¼ bn1 2 2 e A polygonal rough-fuzzy set A in the universe of discourse X can be characterized by n characteristic points a0 ; a1 ; . . . ; an2 e and n characteristic points a0, a1, . . ., an2 and a1 of the lower approximation and an1 of the upper approximation fuzzy set A e as shown in Fig. 5, where A e ¼< ða0 ; a1 ; . . . ; an1 ; l ; l ; . . . ; l Þ; ða0 ; a1 ; . . . ; an1 ; l ; l ; . . . ; l Þ >, the degrees fuzzy set A, 0 1 n1 0 1 n1

l0 ; l1 ; . . . ; ln2 and ln1 , respectively, the degrees of membership of the characteristic points a0, a1, . . ., an2 and an1 are l0, l1, . . ., ln2 and ln1, respectively, and n P 1. In

of membership of the characteristic points a0 ; a1 ; . . . ; an2 and an1 are

Fig. 5, a0 and an1 are called the ‘‘left extreme point’’ and the ‘‘right extreme point’’ of the upper approximation fuzzy set e respectively, a0 and an1 are called the ‘‘left extreme point’’ and the ‘‘right extreme point’’ of the lower approximation A, e respectively, al and ar are called the ‘‘left normal point’’ and the ‘‘right normal point’’ of the upper approximation fuzzy set A, e respectively, al and ar are called the ‘‘left normal point’’ and the ‘‘right normal point’’ of the lower approximation fuzzy set A, e respectively, where n P 1; l ¼ bn1c, and r ¼ dn1e. fuzzy set A, 2 2

A1

A2

...

...

Ak

Ah

...

...

1

μ k,1 μ k,n -2 0

a k,0

a k,1

...

a k,l

a k,r

...

a k,n -2 a k,n -1

X

Fig. 4. Different membership functions for a specific concept defined by different people.

...

...

...

...

...

S.-M. Chen et al. / Information Sciences 299 (2015) 394–411

...

398

...

...

...

e ¼< A; e A e > obtained from the polygonal fuzzy sets shown in Fig. 4. Fig. 5. The constructed polygonal rough-fuzzy set A

e shown in Fig. 5, based on the left normal point al and the right normal point ar , the In the upper approximation fuzzy set A e shown in Fig. 5 can be divided into two parts, i.e., the ‘‘left-hand characteristic points of the upper approximation fuzzy set A side characteristic points’’ and the ‘‘right-hand side characteristic points’’, where the ‘‘left-hand side characteristic points’’ contain the characteristic points (i.e., a0 ; a1 ; . . . ; al Þ which are less than or equal to the left normal point al . The ‘‘right-hand side characteristic points’’ contain the characteristic points (i.e., ar ; . . . ; an2 ; an1 Þ which are larger than or equal to the right e shown in Fig. 5, based on the left normal point al and the right nornormal point ar . In the lower approximation fuzzy set A e shown in Fig. 5 can be divided into two parts, mal point ar, the characteristic points of the lower approximation fuzzy set A i.e., the ‘‘left-hand side characteristic points’’ and the ‘‘right-hand side characteristic points’’, where the ‘‘left-hand side characteristic points’’ contain the characteristic points (i.e., a0,a1, . . ., al) which are less than or equal to the left normal point al. The ‘‘right-hand side characteristic points’’ contain the characteristic points (i.e., ar, . . ., an2,an1) which are larger than or equal to the right normal point ar. e ¼< A; e A e > shown in Fig. 5 from In the following, we propose a method to construct the polygonal rough-fuzzy set A e a set of polygonal fuzzy sets A1, A2, . . ., Ah1 and Ah shown in Fig. 4, where A ¼< ða0 ; a1 ; . . . ; an1 ; l0 ; l1 ; . . . ; ln1 Þ; ða0 ; a1 ; . . . ; an1 ; l0 ; l1 ; . . . ; ln1 Þ >. The proposed method is now presented as follows: e of the polygonal Step 1: Calculate the characteristic points a0 ; a1 ; . . . ; an2 and an1 of the upper approximation fuzzy set A e shown in Fig. 5, where n P 1, shown as follows: rough-fuzzy set A

ag ¼

h X ak;g

!,

h;

ð1Þ

k¼1

e of the polygonal rough-fuzzy set A, e ak, g is the where ag is the gth characteristic point of the upper approximation fuzzy set A gth characteristic point of the kth polygonal fuzzy set Ak shown in Fig. 4, 0 6 g 6 n  1, 1 6 k 6 h, and n P 1. Step 2: Calculate the piecewise fuzzy entropy Ht1,t (Ak) between the t  1th characteristic point and the tth characteristic point of each polygonal fuzzy set Ak shown in Fig. 4, where n P 1, 1 6 t 6 n  1, and 1 6 k 6 h, shown as follows:

Ht1;t ðAk Þ ¼ K

t X

½lk;s logðlk;s Þ þ ð1  lk;s Þ logð1  lk;s Þ;

ð2Þ

s¼t1

where K ¼ 1n ; n is the number of characteristic points of the fuzzy set Ak, lk,s is the degree of membership of the sth characteristic point as belonging to the kth fuzzy set Ak,1 6 t 6 n  1, t  1 6 s 6 t, and 1 6 k 6 h. e between the t  1th charaeristic point and the tth characteristic Step 3: Calculate the piecewise fuzzy entropy Ht1;t A e e shown in Fig. 5, where 1 6 t 6 n  1, shown point of the upper approximation fuzzy set A of the polygonal rough-fuzzy set A as follows:

e ¼ Ht1;t ð AÞ

Ph

k¼1 H t1;t ðAk Þ

h

;

ð3Þ

where Ht1,t(Ak) is the piecewise fuzzy entropy between the t  1th characteristic point and the tth characteristic point of the polygonal fuzzy set Ak, 1 6 t 6 n  1, and 1 6 k 6 h. Step 4: Calculate the degrees of membership l0 ; l1 , . . ., ln2 and ln1 of the characteristic points a0 ; a1 ; . . . ; an2 and an1 e of the polygonal rough-fuzzy set A e shown in Fig. 5, described as follows. belonging to the upper approximation fuzzy set A First, by solving the following non-linear function using the secant method [12], we get the degrees of membership

l0 ; l1 ; . . . ; lu ; . . . ; ll1 and ll of the characteristic points a0 ; a1 ; . . . ; au , . . ., al1 and al belonging to the left-hand side of the e of the polygonal rough-fuzzy set A, e respectively: upper approximation fuzzy set A

         1 e þ ½l log l  þ ð1  l Þ log 1  l ;  lu log lu þ 1  lu log 1  lu ¼ Hu1;u A u1 u1 u1 u1 K

ð4Þ

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  e is the piecewise fuzzy entropy between the u  1th characteristic point and the uth characteristic point of where Hu1;u A e of the polygonal rough-fuzzy set A, e n is the number of characteristic points of the upper the upper approximation fuzzy set A e of the polygonal rough-fuzzy set A; e n P 1; K ¼ 1 ; 1 < u 6 l, and l ¼ bn1c. Then, by solving the folapproximation fuzzy set A n 2 lowing non-linear function using the secant method [12] we get the degrees of membership lr ; lrþ1 , . . ., lv , . . ., ln2 and ln1 of the characteristic points ar ; arþ1 , . . ., av , . . ., an2 and an1 belonging to the right-hand side of the upper approximation fuzzy e of the polygonal rough-fuzzy set A, e respectively: set A

         1     e þ ½l  lv log lv þ 1  lv log 1  lv ¼ Hv ;v þ1 A v þ1 log lv þ1 þ ð1  lv þ1 Þ log 1  lv þ1 ; K

ð5Þ

  e is the piecewise fuzzy entropy between the vth characteristic point and the v + 1th characteristic point of where Hv ;v þ1 A e of the polygonal rough-fuzzy set A, e n is the number of characteristic points of the upper the upper approximation fuzzy set A e of the polygonal rough-fuzzy set A; e n P 1; K ¼ 1 ; r 6 v < n  1, and r ¼ dn1e. approximation fuzzy set A n 2 e e shown in Fig. 5, where the Step 5: Construct the lower approximation fuzzy set A of the polygonal rough-fuzzy set A e is obtained by the intersection of the polygonal fuzzy sets A1, A2, . . ., Ah1 and Ah shown lower approximation fuzzy set A e respecin Fig. 4. Therefore, we can get the characteristic points a0,a1, . . ., an2 and a1 of the lower approximation fuzzy set A, tively, and get the degrees of membership l0,l1, . . ., ln2 and ln1 of the characteristic points a0,a1, . . ., an2 and an1 respectively, where n P 1. e of the polygonal Step 6: Based on the characteristic points a0 ; a1 ; . . . ; an2 and an1 of the upper approximation fuzzy set A e rough-fuzzy set A obtained in Step 1, the degrees of membership l ; l , . . ., l and l of the characteristic points 0

1

n2

n1

e of the polygonal rough-fuzzy set A e obtained in Step a0 ; a1 ; . . . ; an2 and an1 belonging to the upper approximation fuzzy set A e of the polygonal rough-fuzzy set A e obtained in Step 5, we can get the 4, and the constructed lower approximation fuzzy set A e ¼< ða0 ; a1 ; . . . ; an1 ; l ; l ; . . . ; l Þ; ða0 ; a1 ; . . . ; an1 ; l ; l ; . . . ; l Þ >. polygonal rough-fuzzy set A 0 1 n1 0 1 n1 3. Representative value and degree of fuzziness of a polygonal rough-fuzzy set In this section, we propose the definition of the representative value of a polygonal rough-fuzzy set. Moreover, we also propose the definition of the degree of fuzziness of a polygonal rough-fuzzy set. 3.1. Representative value of a polygonal rough-fuzzy set   e of the polygonal rough-fuzzy set In the following, we propose the definition of the representative value Rep A   e e e e e and the repreA ¼< A; A > shown in Fig. 5, where the representative value Rep A of the lower approximation fuzzy set A   e of the upper approximation fuzzy set A e of the polygonal rough-fuzzy set A e are defined as follows: sentative value Rep A

  a þ a þ a þ  þ a 1 2 n1 e ¼ 0 Rep A ; n   a þ a þ a þ  þ a 1 2 n1 e ¼ 0 ; Rep A n

ð6Þ ð7Þ

e of the polygonal roughwhere a0, a1, . . ., an2 and an1 are the characteristic points of the lower approximation fuzzy set A e respectively, a0 ; a1 , . . ., an2 and an1 are the characteristic points of the upper approximation fuzzy set A e of the fuzzy set A,   e e polygonal rough-fuzzy set A, respectively, and n P 1. The representative value Rep A of the polygonal rough-fuzzy set e A e > shown in Fig. 5 is defined as follows: e ¼< A; A

8         >      Rep e A Area e A Rep e A Area e A > >  e  Area A e  > 0   ; if Area A <   >   e e e Area A Area A Rep A ¼ >  >       > >  e ; e  Area A e  ¼ 0 : Rep A if Area A

ð8Þ

    e denotes the area of the lower approximation fuzzy set A e of the polygonal rough-fuzzy set A; e Area A e denotes where Area A     e e e e the area of the upper approximation fuzzy set A of the polygonal rough-fuzzy set A; Area A P 0, and Area A P 0. e1; . . . ; A e z1 ; A e z; A e zþ1 ; . . ., and A e n be polygonal rough-fuzzy sets, where the representative values of Let A e1; . . . ; A e z1 ; A e z; A e zþ1 ; . . ., and A e n are Repð A e 1 Þ; . . . ; Repð A e z1 Þ; Repð A e z Þ; Repð A e zþ1 Þ; . . ., and Repð A e n Þ, respectively, and A e e e e e e e e zþ1 Þ, the polygonal Repð A 1 Þ 6 . . . 6 Repð A z1 Þ 6 Repð A z Þ 6 Repð A zþ1 Þ 6 . . . 6 Repð A n Þ. Because Repð A z1 Þ 6 Repð A z Þ 6 Repð A

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e z1 and the polygonal rough-fuzzy set A e zþ1 are called the left closest polygonal rough-fuzzy set and the rough-fuzzy set A e z , respectively. right closest polygonal rough-fuzzy set of the rough-fuzzy set A 3.2. Degrees of fuzziness of a polygonal rough-fuzzy set In the following, we propose a method to calculate the degree of fuzziness of a polygonal rough-fuzzy set. Based on [19],   e of the polygonal rough-fuzzy set A e ¼< A; e A e > shown in Fig. 5 is defined as follows: the degree of fuzziness Fuzziness A

    e þ Area A e   Area A e ¼ Fuzziness A ; 2

ð9Þ

    e denotes the area of the lower approximation fuzzy set A e of the polygonal rough-fuzzy set A; e Area A e denotes where Area A e the area of the upper approximation fuzzy set A of the polygonal rough-fuzzy set           e e e e e e A; Area A 6 Fuzziness A 6 Area A ; Area A P 0, and Area A P 0. 4. A new fuzzy interpolative reasoning method based on the ratio of fuzziness of the constructed polygonal roughfuzzy sets In this section, we propose a new fuzzy interpolative reasoning method based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets. Let us consider the following fuzzy interpolative reasoning scheme based on polygonal roughfuzzy sets: e 11 and X2 is A e 12 and . . . and Xm is Rule 1: If X1 is A e 21 and X2 is A e 22 and . . . and Xm is Rule 2: If X1 is A .. . e q1 and X2 is A e q2 and . . . and Xm is Rule q: If X1 is A

e 1m Then Y is B e1 A e 2m Then Y is B e2 A

e qm Then Y is B eq A    e e e Observation: X1 is A 1 and X2 is A 2 and . . . and Xm is A m e Conclusion: Y is B

where Xj denotes the jth antecedent variable; Y denotes the consequence variable; the values of the antecedent variables X1, X2, . . ., and Xm and the values of the consequence variable Y appearing in the fuzzy rules are polygonal rough-fuzzy sets constructed by the proposed polygonal rough-fuzzy sets construction method presented in Section 2.2; the antecedent polyge ij of Rule i is represented by A e ij ¼< ðaij;0 , aij,1, . . ., aij,n1; lij,0, lij,1, . . ., lij,n1), onal rough-fuzzy set A e ij of the antecedent polygonal (aij;0 ; aij;1 ; . . . ; aij;n1 ; lij;0 ; lij;1 , . . ., lij;n1 Þ >, where the upper approximation fuzzy set A e ij is characterized by n characteristic points aij;0 ; aij;1 , . . ., aij;n2 and aij;n1 , and l ; l , . . ., l rough-fuzzy set A ij;0 ij;1 ij;n2 and lij;n1 are the degrees of membership of the characteristic points aij;0 ; aij;1 , . . ., aij;n2 and aij;n1 belonging to the upper approxe ij of the antecedent polygonal rough-fuzzy set A e ij , respectively. The lower approximation fuzzy set A e ij of imation fuzzy set A e ij is characterized by n characteristic points aij,0, aij,1, . . ., aij, the antecedent polygonal rough-fuzzy set A

lij,0, lij,1, . . ., lij,

and lij,

n2

and aij,

n1,

and

are the degrees of membership of the characteristic points aij,0, aij,1, . . ., aij, n2 and aij, n1 e ij of the antecedent polygonal rough-fuzzy set A e ij , respectively, 1 6 i 6 q, belonging to the lower approximation fuzzy set A n2

n1

e  is represented by A e  ¼< ða ; a , . . ., a ; l ; l , . . ., 1 6 j 6 m, and n P 1. The observation polygonal rough-fuzzy set A j j j;0 j;1 j;n1 j;0 j;1

lj;n1 Þ, (aj;0 ; aj;1 , . . ., aj;n1 ; lj;0 ; lj;1 , . . ., lj;n1 Þ >, where the upper approximation fuzzy set Ae j of the observation polygonal  e  is characterized by n characteristic points a ; a , . . ., a rough-fuzzy set A j j;0 j;1 j;n2 and aj;n1 , and

lj;0 ; lj;1 , . . ., lj;n2 and lj;n1

are the degrees of membership of the characteristic points aj;0 ; aj;1 , . . ., aj;n2 and aj;n1 belonging to the upper approximation e  of the observation polygonal rough-fuzzy set A e  , respectively. The lower approximation fuzzy set A e  of the fuzzy set A j j j  e  is characterized by n characteristic points a ; a , . . ., a observation polygonal rough-fuzzy set A j j;0 j;1 j;n2 and aj;n1 , and

lj;0 ; lj;1 , . . ., lj;n2 and lj;n1 are the degrees of membership of the characteristic points aj;0 ; aj;1 , . . ., aj;n2 and aj;n1 belonging e  of the observation polygonal rough-fuzzy set A e  , respectively, 1 6 j 6 m, and n P 1; to the lower approximation fuzzy set A j j e i of each fuzzy rule Rule i is represented by B e i ¼< ðbi;0 , bi,1, . . ., bi, the consequence polygonal rough-fuzzy set B

n1;

li,0, li,1,

e i of the consequence polygonal . . ., li, n1), (bi;0 ; bi;1 ; . . . ; bi;n1 ; li;0 ; li;1 , . . ., li;n1 Þ >, where the upper approximation fuzzy set B

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e i is characterized by n characteristic points bi;0 ; bi;1 , . . ., bi;n2 and bi;n1 , and l ; l , . . ., l rough-fuzzy set B i;0 i;1 i;n2 and li;n1 are the degrees of membership of the characteristic points bi;0 ; bi;1 , . . ., bi;n2 and bi;n1 belonging to the upper approximation e i of the consequence polygonal rough-fuzzy set B e i , respectively. The lower approximation fuzzy set B e i of the confuzzy set B e i is characterized by n characteristic points bi,0, bi,1, . . ., bi, n2 and bi, n1, and li,0, li,1, . . ., sequence polygonal rough-fuzzy set B

li, n2 and li, n1 are the degrees of membership of the characteristic points bi,0, bi,1, . . ., bi, n2 and bi, n1 belonging to the e i of the consequence polygonal rough-fuzzy set B e i , respectively, 1 6 i 6 q, and n P 1; the lower approximation fuzzy set B  e e  ¼< ðb ; b , . . ., b ; l ; l , . . ., fuzzy interpolative reasoning result B is a polygonal rough-fuzzy set represented by B 0 1 0 1 n1

ln1 Þ, (b0 ; b1 ; . . . ; bn1 ; l0 ; l1 , . . ., ln1 Þ >, where the upper approximation fuzzy set Be  of the fuzzy interpolative reasoning e  represented by a polygonal rough-fuzzy set is characterized by n characteristic points b ; b , . . ., b and b , result B 0 1 n2 n1 and

l0 ; l1 , . . ., ln2 and ln1 are the degrees of membership of the characteristic points b0 ; b1 ; . . . ; bn2 and bn1 belonging

e  of the fuzzy interpolative reasoning result B e  represented by a polygonal roughto the upper approximation fuzzy set B  e e  represented fuzzy set, respectively. The lower approximation fuzzy set B of the fuzzy interpolative reasoning result B 







by a polygonal rough-fuzzy set is characterized by n characteristic points b0 ; b1 ; . . . ; bn2 and bn1 , and

l

 n1

are the degrees of membership of the characteristic points

   b0 ; b1 ; . . . ; bn2 and

 bn1

l0 ; l1 ; . . . ; ln2 and

belonging to the lower approximation

e  of the fuzzy interpolative reasoning result B e  represented by a polygonal rough-fuzzy set, respectively, and fuzzy set B n P 1; m denotes the number of antecedent variables appearing in the fuzzy rules; q denotes the number of fuzzy rules. The proposed fuzzy interpolative reasoning method based on the ratio of fuzziness of the constructed polygonal roughfuzzy sets is now presented as follows: Step 1: Calculate the left placement factor clj and the right placement factor crj of each observation polygonal rough-fuzzy e  , respectively, where 1 6 j 6 m, shown as follows: set A j

clj ¼

crj ¼

8 > > > <

 

1

> > > : 1; 8 > > > <

Rep e A j Repðe A lj Þ

      e lj 6 Rep A e  6 Rep A e rj   ; if Rep A j A rj Repðe A lj Þ Rep e     e ij < Rep A e  ; where 1 6 i 6 q if Rep A j  

1

> > > : 1;

ð10Þ

 

Rep e A rj Rep e A j

      e lj 6 Rep A e  6 Rep A e rj   ; if Rep A j Rep e A rj Repðe A lj Þ     e  < Rep A e ij ; where 1 6 i 6 q if Rep A

ð11Þ

j

    e ij denotes the representative value of the antecedent polygonal rough-fuzzy set A e ij ; Rep A e  denotes the repwhere Rep A j  e e resentative value of the observation polygonal rough-fuzzy set A ; A lj is the left closest polygonal rough-fuzzy set of the j

e; A e rj is the right closest polygonal rough-fuzzy set of the observation polygonal observation polygonal rough-fuzzy set A j  e rough-fuzzy set A j , where 1 6 i 6 q and 1 6 j 6 m. Step 2: Based on [19], calculate the weight Wi of each fuzzy rule Rule i, where 1 6 i 6 q, shown as follows:

minj¼1;2;...;m cij W i ¼ Ph ; i¼1 minj¼1;2;...;m cij

ð12Þ

P where 1 6 i 6 q, 1 6 j 6 m, 0 6 cij 6 1, 0 6 minj=1,2,. . ., mcij 6 1, 0 6 Wi 6 1, and qi¼1 W i ¼ 1. e 0 . Firstly, calculate the characteristic points b0 ; b0 ; . . . ; b0 Step 3: Get the intermediate polygonal rough-fuzzy set B 0 1 n2 and 0 0 0 e e b of the lower approximation fuzzy set B of the intermediate polygonal rough-fuzzy set B and calculate the degrees n1

0

0

0

0

of membership l00 ; l01 ; . . . ; l0n2 and l0n1 of the characteristic points b0 ; b1 ; . . . ; bn2 and bn1 of the lower approximation fuzzy e 0 of the intermediate polygonal rough-fuzzy set B e 0 , respectively, where n P 1, shown as follows: set B 0

by ¼

l0y ¼

q X W i  bi;y ; i¼1 q X

W i  li;y ;

ð13Þ ð14Þ

i¼1

P 0 0 0 0 where 1 6 i 6 q, 0 6 y 6 n  1, 0 6 Wi 6 1, and qi¼1 W i ¼ 1. Then, calculate the characteristic points b0 ; b1 ; . . . ; bn2 and bn1 e 0 of the intermediate polygonal rough-fuzzy set B e 0 and calculate the degrees of memof the upper approximation fuzzy set B

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0 0 0 0 e0 bership l00 ; l01 ; . . . ; l0n2 and l0n1 of the characteristic points b0 ; b1 ; . . . ; bn2 and bn1 of the upper approximation fuzzy set B 0 e of the intermediate polygonal rough-fuzzy set B , respectively, where n P 1, shown as follows:

0

by ¼

l0y ¼

q X W i  bi;y ;

ð15Þ

i¼1 q X

W i  li;y ;

ð16Þ

i¼1

Pq where 1 6 i 6 q, 0 6 y 6 n-1, 0 6 Wi6 1, and i¼1 W i ¼ 1. Then, we can get the intermediate polygonal rough-fuzzy set e 0 ¼< ðb0 ; b0 ; . . . ; b0 ; l0 ; l0 ; . . . ; l0 Þ, (b0 ; b0 ; . . . ; b0 ; l0 ; l0 ; . . . ; l0 Þ >. B 0 1 n1 0 1 n1 0 1 n1 0 1 n1   e ; A e ; . . . ; A e  of the observation polygonal roughStep 4: Calculate the average degree of fuzziness Average_Fuzziness A 1

2

m

e ; A e ; . . . ; A e e e e e fuzzy sets A 1 2 m1 and A m and calculate the average degree of fuzziness Average_Fuzziness( A i1 ; A i2 , . . ., A im ) of the e i1 ; A e i2 , . . ., A e iðm1Þ and A e im of the fuzzy rule Rule i, respectively, where 1 6 i 6 q antecedent polygonal rough-fuzzy sets A and 1 6 j 6 m, shown as follows:

  e   Fuzziness A j    m e ;A e ;...;A e Av erage Fuzziness A ; ¼ um 1 2 m j¼1 m m e ij Þ X Fuzzinessð A e i1 ; A e i2 ; . . . ; A e im Þ ¼ ; Av erage Fuzzinessð A m j¼1

ð17Þ ð18Þ

where 1 6 i 6 q and 1 6 j 6 m. Step 5: Calculate the ratio R of the average degree of fuzziness of the observation polygonal rough-fuzzy sets to the average degree of fuzziness of the antecedent polygonal rough-fuzzy sets of the q fuzzy rules, shown as follows:

R ¼ Pq

  e ; A e ; . . . ; A e Av erage Fuzziness A 1 2 m

i¼1 W i

e i1 ; A e i2 ; . . . ; A e im Þ  Av erage Fuzzinessð A

;

ð19Þ

P where R P 0, Wi is the weight of the fuzzy rule Rule i, 1 6 i 6 q, 0 6 Wi 6 1, and qi¼1 W i ¼ 1.  e represented by a polygonal rough-fuzzy set based on the following Step 6: Get the fuzzy interpolative reasoning result B     e  of the calculations. Firstly, calculate the characteristic points b0 ; b1 ; . . . ; bn2 and bn1 of the lower approximation fuzzy set B e  represented by a polygonal rough-fuzzy set, respectively, where n P 1, shown as fuzzy interpolative reasoning result B follows:

  0 0 0 bc ¼ bl þ br =2; 

0



0

0 br

2 0  bl

bl ¼ bc  R  

ð20Þ

0 br

0 bl

;

br ¼ bc þ R  ; 2 8 l1   X > >  0 0 > > bl  R  bqþ1  bq ; if 0 6 s 6 l  1 > < q¼s  bs ¼ s1 > X > 0 0 > > br þ R  ðbqþ1  bq Þ; if r þ 1 6 s 6 n  1 > :

ð21Þ ð22Þ

ð23Þ

q¼r

    e where 0 6 s 6 n  1. Then, calculate the characteristic points b0 ; b1 , . . ., bn2 and bn1 of the upper approximation fuzzy set B  e represented by a polygonal rough-fuzzy set, respectively, where n P 1, shown of the fuzzy interpolative reasoning result B

as follows:

  0 0 0 bc ¼ bl þ br =2; 0

ð24Þ 0

br  bl ; 2 0 0 b  bl  0 br ¼ bc þ R  r ; 2 8 l1   X > >  0 0 > > bl  R  bqþ1  bq ; if 0 6 s 6 l  1 > < q¼s  bs ¼ s1 > X >  0 0 >b þ > R  ðbqþ1  bq Þ; if r þ 1 6 s 6 n  1 > : r 

0

bl ¼ bc  R 

q¼r

ð25Þ ð26Þ

ð27Þ

S.-M. Chen et al. / Information Sciences 299 (2015) 394–411

403

Fig. 6. Fuzzy interpolative reasoning result of Example 5.1 based on the proposed method with multiple fuzzy rules.

where 0 6 s 6 n  1. Finally, calculate the degrees of membership    b0 ; b1 ; . . . ; bn2

and

 bn1

l0 ; l1 ; . . . ; ln2 and ln1 of the characteristic points

e  of the fuzzy interpolative reasoning result B e  repbelonging to the lower approximation fuzzy set B

resented by a polygonal rough-fuzzy set and calculate the degrees of membership l0 ; l1 , . . ., ln2 and ln1 of the character    e  of the fuzzy interpolative reasoning istic points b0 ; b1 ; . . . ; bn2 and bn1 belonging to the upper approximation fuzzy set B  e result B represented by a polygonal rough-fuzzy set, respectively, where n P 1, shown as follows:

ly ¼ ly ¼

q X W i  li;y ; i¼1 q X

W i  li;y ;

ð28Þ ð29Þ

i¼1

P where 1 6 i 6 q, 0 6 y 6 n  1, 0 6 Wi 6 1, and qi¼1 W i ¼ 1. Thus, we can get the of the fuzzy interpolative reasoning result e  represented by a polygonal rough-fuzzy set, where B e  ¼< ðb ; b ; . . . ; b ; l ; l ; . . . ; l Þ, (b ; b , . . ., b ; l ; l , . . ., B 0 1 n1 0 1 n1 0 1 n1 0 1

ln1 >). 5. Experimental results In this section, we use some examples to compare the experimental results of the proposed fuzzy interpolative reasoning method with the ones of Chen and Shen’s method [13]. Example 5.1. Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are polygonal rough-fuzzy sets, as shown in Fig. 6. The fuzzy interpolative reasoning scheme based on polygonal rough-fuzzy sets with multiple fuzzy rules is shown as follows: e 1 Then Y is B e1 Rule 1: If X1 is A e 2 Then Y is B e2 Rule 2: If X1 is A e e3 Rule 3: If X1 is A 3 Then Y is B  e Observation: X1 is A e Conclusion: Y is B where

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e 1 ¼< ð2; 3; 3:5; 4:5; 5; 6; 0; 0:6; 0:65; 0:7; 0:6; 0Þ A ð1:5; 2:5; 3; 5:5; 6; 6:5; 0; 0:9; 1; 1; 0:85; 0Þ >; e A 2 ¼< ð8; 8:5; 9; 10; 10:5; 11; 0; 0:4; 0:5; 0:6; 0:5; 0Þ; ð7; 8:5; 9; 10; 11; 12:5; 0; 0:7; 1; 1; 0:9; 0Þ >; e A 3 ¼< ð20; 21; 21:5; 22; 22:5; 23; 0; 0:5; 0:55; 0:6; 0:55; 0Þ; ð19:5; 20:5; 21; 22; 23; 23:5; 0; 0:8; 1; 1; 0:9; 0Þ >;  e A ¼< ð14:5; 15; 15:5; 16; 16:5; 17:5; 0; 0:5; 0:6; 0:7; 0:6; 0Þ; ð13; 14:5; 15; 16; 17:5; 18:5; 0; 0:8; 1; 1; 0:9; 0Þ >; e B 1 ¼< ð1:5; 2; 3; 4; 5; 5:5; 0; 0:3; 0:6; 0:5; 0:4; 0Þ; ð1; 2; 3; 4; 5; 6; 0; 0:7; 1; 1; 0:6; 0Þ >; e 2 ¼< ð8; 8:5; 9; 10; 10:5; 11; 0; 0:55; 0:65; 0:6; 0:5; 0Þ; B ð7:5; 8:5; 9; 10; 11; 12:5; 0; 0:8; 1; 1; 0:9; 0Þ > : e 3 ¼< ð19:5; 20; 21; 21:5; 22; 22:5; 0; 0:6; 0:7; 0:65; 0:55; 0Þ; B ð19; 19:5; 20; 21:5; 22; 23; 0; 0:8; 1; 1; 0:85; 0Þ > : e 1 Þ; Repð A e 2 Þ; Repð A e  Þ and Repð A e 3 Þ of the lower approximations Based on Eq. (6), we can get the representative values Repð A   e e e e e e e e e 1 Þ ¼ 2þ3þ3:5þ4:5þ5þ6 = 4, A 1 ; A 2 ; A , and A 3 of the polygonal rough-fuzzy sets A 1 ; A 2 ; A and A 3 , respectively, where Repð A 6 8þ8:5þ9þ10þ10:5þ11  14:5þ15þ15:5þ16þ16:5þ17:5 20þ21þ21:5þ22þ22:5þ23 e e e Repð A 2 Þ ¼ = 9.5, Repð A Þ ¼ = 15.83 and Repð A 3 Þ ¼ = 21.67. Based on 6

6

6

e 1 Þ; Repð A e 2 Þ; Repð A e  Þ, and Repð A e 3 Þ of the upper approximations A e1; A e2; A e, Eq. (7), we can get the representative values Repð A e 3 of the polygonal rough-fuzzy sets A e1; A e2; A e  , and A e 3 , respectively, where Repð A e 1 Þ ¼ 1:5þ2:5þ3þ5:5þ6þ6:5 = 4.17, and A 6 e 2 Þ ¼ 7þ8:5þ9þ10þ11þ12:5 = 9.67, Repð A e  Þ ¼ 13þ14:5þ15þ16þ17:5þ18:5 ¼ 15:75, and Repð A e 3 Þ ¼ 19:5þ20:5þ21þ22þ23þ23:5 ¼ 21:58. Based Repð A 6 6 6 e 1 Þ; Repð A e 2 Þ; Repð A e  Þ, and Repð A e 3 Þ of the polygonal rough-fuzzy sets on Eq. (8), we can get the representative values Repð A e2; A e  , and A e 3 , respectively, where e1; A A

        e 1  Area A e 1  Rep A e 1  Area A e1 Rep A 4:17  4:13  4  1:88 e1Þ ¼ Repð A ¼ ¼ 4:31;     4:13  1:88 e 1  Area A e1 Area A         e 2  Area A e 2  Rep A e 2  Area A e2 Rep A 9:67  3:33  9:5  1:15 e2Þ ¼ Repð A ¼ ¼ 9:76;     3:33  1:15 e e Area A 2  Area A 2         e   Area A e   Rep A e   Area A e Rep A 15:75  3:8  15:83  1:23 eÞ ¼ Repð A ¼ ¼ 15:71;     3:8  1:23 e   Area A e Area A         e 3  Area A e 3  Rep A e 3  Area A e3 Rep A 21:58  2:95  21:67  1:18 e3Þ ¼ Repð A ¼ ¼ 21:52:     2:95  1:18 e e Area A 3  Area A 3 e 1 Þ < Repð A e 2 Þ < Repð A e  Þ < Repð A e 3 Þ, we can see that the antecedent polygonal rough-fuzzy set A e 2 and the anteBecause Repð A e 3 are the left closest polygonal rough-fuzzy set and the right closest polygonal roughcedent polygonal rough-fuzzy set A e  , respectively. From Fig. 6, we can see that the fuzzy interpolative fuzzy set of the observation polygonal rough-fuzzy set A e  of the proposed method represented by a polygonal rough-fuzzy set is B e  = <(13.54, 14.13, 15, 15.88, 16.47, 17.05; 0, result B 0.58, 0.68, 0.63, 0.53, 0), (13, 13.87, 14.46, 15.92, 16.8, 18.25; 0, 0.8, 1, 1, 0.87, 0)>. From the experimental result shown in Fig. 6, we can see that the proposed fuzzy interpolative reasoning method can overcome the drawback of Chen and Shen’s method [13], which cannot deal with fuzzy interpolative reasoning using polygonal rough-fuzzy sets in this situation. Example 5.2. [13]: Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are triangular rough-fuzzy sets, as shown in Fig. 7. The fuzzy interpolative reasoning scheme based on triangular rough-fuzzy sets is shown as follows:

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405

Fig. 7. A comparison of fuzzy interpolative reasoning results of Example 5.2 for different methods based on triangular rough-fuzzy sets.

e 1 Then Y is B e1 Rule 1: If X1 is A e 2 Then Y is B e2 Rule 2: If X1 is A  e Observation: X1 is A e Conclusion: Y is B where

e 1 ¼< ð1; 3:5; 4; 0; 0:7; 0Þ; ð0; 4; 5; 0; 1; 0Þ >; A e 2 ¼< ð12; 13; 13:5; 0; 0:7; 0Þ; ð11; 13; 14; 0; 1; 0Þ >; A e  ¼< ð6:5; 8; 9:5; 0; 0:6; 0Þ; ð6; 8; 10; 0; 1; 0Þ >; A e 1 ¼< ð1:5; 2; 3; 0; 0:5; 0Þ; ð0; 2; 4; 0; 1; 0Þ >; B e 2 ¼< ð11; 11:5; 12; 0; 0:5; 0Þ; ð10; 11; 13; 0; 1; 0Þ > : B e  of Chen and Shen’s method [13] represented by a triangular From Fig. 7, we can see that the fuzzy interpolative result B e  ¼< ð6:28; 6:70; 7:95; 0; 0:43; 0Þ; ð5:31; 6:28; 8:83; 0; 1; 0Þ > and the fuzzy interpolative result B e  of rough-fuzzy set is B  e the proposed method represented by a triangular rough-fuzzy set is B ¼< ð6:37; 7:02; 7:67; 0; 0:5; 0Þ; ð5:07; 6:90; 8:73; 0; 1; 0Þ >. Therefore, we can see that the fuzzy interpolative result of the proposed method is more reasonable than Chen and Shen’s method [13] in terms of the shapes of the upper approximation fuzzy set and the lower approxe  with respect to the given triangular rough-fuzzy sets imation fuzzy set of the fuzzy interpolative reasoning result B e 2; B e . e1; A e1; B e 2 , and A A Example 5.3. [13]: Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are triangular rough-fuzzy sets, as shown in Fig. 8. The fuzzy interpolative reasoning scheme based on triangular rough-fuzzy sets is shown as follows:

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Fig. 8. A comparison of fuzzy interpolative reasoning results of Example 5.3 for different methods based on triangular rough-fuzzy sets.

e 1 Then Y is B e1 Rule 1: If X1 is A e 2 Then Y is B e2 Rule 2: If X1 is A  e Observation: X1 is A e Conclusion: Yis B where

e 1 ¼< ð1; 3:5; 4; 0; 0:7; 0Þ; ð0; 4; 5; 0; 1; 0Þ >; A e 2 ¼< ð12; 13; 13:5; 0; 0:7; 0Þ; ð11; 13; 14; 0; 1; 0Þ >; A e  ¼< ð7; 8; 9; 0; 0:6; 0Þ; ð6; 8; 10; 0; 1; 0Þ >; A e 1 ¼< ð1:5; 2; 3; 0; 0:5; 0Þ; ð0; 2; 4; 0; 1; 0Þ >; B e 2 ¼< ð11; 11:5; 12; 0; 0:5; 0Þ; ð10; 11; 13; 0; 1; 0Þ > : B e  of Chen and Shen’s method [8] represented by a triangular From Fig. 8, we can see that the fuzzy interpolative result B e  ¼< ð6:52; 6:81; 7:64; 0; 0:43; 0Þ; ð5:32; 6:31; 8:83; 0; 1; 0Þ > and the fuzzy interpolative result B e rough-fuzzy set is B  e of the proposed method represented by a triangular rough-fuzzy set is B ¼< ð6:44; 7:02; 7:61; 0; 0:5; 0Þ; ð5:26; 6:90; 8:54; 0; 1; 0Þ >. Therefore, the fuzzy interpolative result of the proposed method is more reasonable than Chen and Shen’s method [13] in terms of the shapes of the upper approximation fuzzy set and the lower approximation fuzzy e 1; A e2; B e. e  with respect to the given triangular rough-fuzzy sets A e1; B e 2 , and A set of the fuzzy interpolative reasoning result B Example 5.4. : Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are triangular rough-fuzzy sets, as shown in Fig. 9. The fuzzy interpolative reasoning scheme based on triangular rough-fuzzy sets is shown as follows:

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407

Fig. 9. A comparison of fuzzy interpolative reasoning results of Example 5.4 for different methods based on triangular rough-fuzzy sets.

e 1 Then Y is B e1 Rule 1: If X1 is A e e2 Rule 2: If X1 is A 2 Then Y is B  e Observation: X1 is A e Conclusion: Y is B where

e 1 ¼< ð3; 3; 3; 0; 1; 0Þ; ð3; 3; 3; 0; 1; 0Þ >; A e 2 ¼< ð12; 13; 13:5; 0; 0:6; 0Þ; ð11; 13; 14; 0; 1; 0Þ >; A e  ¼< ð6; 7; 8; 0; 0:6; 0Þ; ð5; 7; 9; 0; 1; 0Þ >; A e 1 ¼< ð4; 4; 4; 0; 1; 0Þ; ð4; 4; 4; 0; 1; 0Þ >; B e 2 ¼< ð10:5; 11:5; 12; 0; 0:5; 0Þ; ð10; 11:5; 13; 0; 1; 0Þ > : B e  of Chen and Shen’s method [13] represented by a triangular From Fig. 9, we can see that the fuzzy interpolative result B e  ¼< ð5:98; 7:04; 7:98; 0; 0:57; 0Þ; ð5:27; 6:66; 9:27; 0; 1; 0Þ > and the fuzzy interpolative result B e  of the rough-fuzzy set is B  e proposed method represented by a triangular rough-fuzzy set is B ¼< ð6:02; 7:02; 8:02; 0; 0:79; 0Þ; ð5:13; 7:13; 9:13; 0; 1; 0Þ >. Therefore, the fuzzy interpolative result of the proposed method is more reasonable than Chen and Shen’s method [13] in terms of the shapes of the upper approximation fuzzy set and the lower approximation fuzzy set of the fuzzy intere1; A e 2; B e . e  with respect to the given triangular rough-fuzzy sets A e1; B e 2 , and A polative reasoning result B Example 5.5. [13]: Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are triangular type-1 fuzzy sets, which are represented by triangular rough-fuzzy sets, as shown in Fig. 10, where the fuzzy interpolative reasoning scheme is shown as follows:

408

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Fig. 10. A comparison of fuzzy interpolative reasoning results of Example 5.5 for different methods based on triangular type-1 fuzzy sets.

e 1 Then Y is B e1 Rule 1: If X1 is A e e2 Rule 2: If X1 is A 2 Then Y is B e Observation: X1 is A e Conclusion: Y is B where

e 1 ¼< ð0; 5; 6; 0; 1; 0Þ; ð0; 5; 6; 0; 1; 0Þ >; A e 2 ¼< ð11; 13; 14; 0; 1; 0Þ; ð11; 13; 14; 0; 1; 0Þ >; A e  ¼< ð7; 8; 9; 0; 1; 0Þ; ð7; 8; 9; 0; 1; 0Þ >; A e 1 ¼< ð0; 2; 4; 0; 1; 0Þ; ð0; 2; 4; 0; 1; 0Þ >; B e 2 ¼< ð10; 11; 13; 0; 1; 0Þ; ð10; 11; 13; 0; 1; 0Þ > : B e  of Chen and Shen’s method [13] represented by a triangular From Fig. 10, we can see that the fuzzy interpolative result B  e ¼<(5.83, 6.26, 7.38; 0, 1, 0), (5.83, 6.26, 7.38; 0, 1, 0)> and the fuzzy interpolative result B e  of the prorough-fuzzy set is B e  = <(5.80, 6.57, 7.35; 0, 1, 0), (5.80, 6.57, 7.35; 0, 1, 0)>. Thereposed method represented by a triangular rough-fuzzy set is B fore, the fuzzy interpolative result of the proposed method is more reasonable than Chen and Shen’s method [13] in terms of e  with respect to the given triangular the shape of the membership function of the fuzzy interpolative reasoning result B e1; A e2; B e  represented by triangular rough-fuzzy sets. e1; B e 2 , and A type-1 fuzzy sets A Example 5.6. Let us consider the situation that the antecedents and the consequences of the given fuzzy rules and the observation are polygonal rough-fuzzy sets, as shown in Fig. 11. The fuzzy interpolative reasoning scheme based on polygonal rough-fuzzy sets is shown as follows:

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409

Fig. 11. Fuzzy interpolative reasoning result of Example 5.6 based on the proposed method.

e 1 Then Y is B e1 Rule 1: If X1 is A e e2 Rule 2: If X1 is A 2 Then Y is B  e Observation: X1 is A e Conclusion: Y is B

where

e 1 ¼< ð1:5; 2; 3; 4; 5; 5:5; 0; 0:2; 0:6; 0:5; 0:3; 0Þ; A ð1; 2; 3; 4; 5; 6; 0; 0:6; 1; 1; 0:7; 0Þ >; e A 2 ¼< ð14:5; 15; 15:5; 16; 16:5; 17:5; 0; 0:6; 0:65; 0:7; 0:5; 0Þ; ð13; 14:5; 15; 16; 17:5; 18:5; 0; 0:9; 1; 1; 0:8; 0Þ >;  e A ¼< ð8; 8:5; 9; 10; 10:5; 11; 0; 0:5; 0:6; 0:55; 0:5; 0Þ; ð7; 8:5; 9; 10; 11; 12:5; 0; 0:8; 1; 1; 0:8; 0Þ >; e 1 ¼< ð3; 3:5; 4; 4:5; 6; 6:5; 0; 0:5; 0:6; 0:6; 0:45; 0Þ; B ð2:5; 3:5; 4; 5:5; 6; 7; 0; 0:8; 1; 1; 0:9; 0Þ >; e 2 ¼< ð16:5; 17; 17:5; 18; 18:5; 19; 0; 0:5; 0:55; 0:6; 0:45; 0Þ; B ð15; 16:5; 17:5; 18:5; 19; 19:5; 0; 0:9; 1; 1; 0:85; 0Þ > : e  of the proposed method represented by a polygonal roughFrom Fig. 11, we can see that the fuzzy interpolative result B e  ¼< ð9:97; 10:46; 10:95; 11:43; 12:4; 12:88; 0; 0:5; 0:57; 0:6; 0:45; 0Þ; ð8:99; 10:22; 10:96; 12:17; 12:66; 13:38; fuzzy set is B 0; 0:85; 1; 1; 0:87; 0Þ >. From the experiment result, we can see that the proposed fuzzy interpolative reasoning method can overcome the drawback of Chen and Shen’s method [13], which cannot deal with polygonal rough-fuzzy sets for fuzzy interpolative reasoning in this situation.

6. Conclusions In this paper, we have proposed a method to construct a polygonal rough-fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts’ opinions and have proposed a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets. From the

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experimental results shown in Section 5, we can see that the proposed fuzzy interpolative reasoning method can overcome the drawbacks of Chen and Shen’s method [13] for fuzzy interpolative reasoning in sparse fuzzy rule-based systems based on the ratio of fuzziness of the constructed polygonal rough-fuzzy sets. Chen and Shen [13] have pointed out that rough-fuzzy sets and type-2 fuzzy sets are different for the following two aspects: (1) The membership function of a type-2 fuzzy set is defined in a three-dimensional space, whereas the membership function of a rough-fuzzy set is defined in a two-dimensional space and (2) A type-2 fuzzy set uses type-1 fuzzy sets [68] to represent the membership values of the elements, whereas the concept of rough-fuzzy sets is based on the definition of rough sets [54], where two regions, referred to as the lower and upper approximations, are used to capture the uncertainty. Moreover, Chen and Shen [13] also pointed out that the representation of rough-fuzzy sets is different from the representation of type-2 fuzzy sets. Therefore, because rough-fuzzy sets and type-2 fuzzy sets are different for the above two aspects and because the representation of rough-fuzzy sets is different from the representation of type-2 fuzzy sets, in this paper, we do not compare the experimental results of the proposed fuzzy interpolative reasoning method with the ones of the existing fuzzy interpolative reasoning methods based on type-2 fuzzy sets or interval type-2 fuzzy sets, where we focus on the comparison of the experimental results of the proposed fuzzy interpolative reasoning method with Chen and Shen’s method [13] based on rough-fuzzy sets and type-1 fuzzy sets. In the future, we will propose a fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on interval type-2 roughfuzzy sets [69], where we will compare the experimental results of the proposed fuzzy interpolative reasoning method based on interval type-2 rough-fuzzy sets with the ones of the existing fuzzy interpolative reasoning methods based on type-2 fuzzy sets or interval type-2 fuzzy sets.

Acknowledgement This work was supported in part by the Ministry of Science and Technology, Republic of China, under Grant MOST 1032221-E-011-108-MY2.

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