Some notes on optimal fuzzy reasoning methods

Information Sciences 503 (2019) 652–669

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

Some notes on optimal fuzzy reasoning methods Yingfang Li a, Xingxing He b,∗, Keyun Qin b, Dan Meng a a b

School of Economics Information Engineering, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

a r t i c l e

i n f o

Article history: Received 8 October 2016 Revised 15 May 2018 Accepted 2 July 2019 Available online 3 July 2019 Keywords: Optimal fuzzy reasoning Compositional rule of inference Pointwise optimizing fuzzy inference Fuzzy implication Fuzzy reasoning

a b s t r a c t The optimal fuzzy reasoning (OFR) method proposed by Zhang and Cai (2004) is treated as a control problem and it is dramatically different from other existing fuzzy reasoning methods. The aim of this paper is to make some notes on OFR methods. We give a modification of an OFR method presented by Zhang and Cai. Some advantages and limitations of OFR methods are examined. We introduce several new OFR methods based on the Łukasiewicz implication, the Kleene-Dienes implication and the Zadeh implication. The reasoning consistency of the proposed methods is studied. The relationship between the OFR method and other fuzzy reasoning methods (e.g., the compositional rule of inference method and the pointwise optimizing fuzzy inference method) is examined. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The cognitive process of human reasoning deals with imprecise information, which is often represented by fuzzy sets or fuzzy relations. In order to tackle the imprecise information in human reasoning, Zadeh developed fuzzy reasoning methodology, which allows us to derive an approximate conclusion from imprecise knowledge [20–22]. The core research topics on fuzzy reasoning are fuzzy reasoning methods and their analysis. Various methods have been presented for fuzzy reasoning, including the compositional rule of inference (CRI) method [20–22], the evidence reasoning method [5,7], the interpolative reasoning method [1,6], the similarity based approximate reasoning method [12,15–17], the triple implication method [18,19], and so on. These existing fuzzy reasoning methods are essentially static mappings from the universe of single fuzzy premises to the universe of single fuzzy consequences. They are often treated as logical inference processes and have a strong relation to the classical logical inference. Thus the corresponding logic foundation of them is not very clear yet. In [4,23], Zhang and Cai proposed a new fuzzy reasoning method called the optimal fuzzy reasoning (OFR) method, which maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of underlying reasoning goals. In this way, fuzzy reasoning is no longer a logical inference but a process of computation and optimization. Each OFR method is treated as a control problem and determined by a reasoning goal specified by a given objective function. Given a set of fuzzy rules, an identical fuzzy premise implies different fuzzy consequences for different reasoning goals. This makes OFR methods dramatically different from other existing fuzzy reasoning methods. From the control perspective, the processes of mostly existing fuzzy reasoning methods are of open-loop but OFR methods are of closed-loop. Owing to the closed-loop characteristic, OFR methods can significantly improve robustness and consistency [9]. The motivation of this paper is to make some notes on OFR methods. ∗

Corresponding author. E-mail address: [email protected] (X. He).

https://doi.org/10.1016/j.ins.2019.07.013 0020-0255/© 2019 Elsevier Inc. All rights reserved.

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

653

Suppose the fuzzy premise is given as [ai ]1 × m , which implies a fuzzy consequence denoted as [bj ]1 × n . The reasoning goals of OFR methods are to minimize some objective functions such as

Jop =

m  n 

|r (i, j ) − min(ai , b j )|,

i=1 j=1

 Kop =

m  n 

(r (i, j ) − min(ai , b j ))2 ,

i=1 j=1

Lop =

max

1≤i≤m,1≤ j≤n

|r (i, j ) − min(ai , b j )|.

The objective functions Jop , Kop and Lop are used to measure the absolute distances between the given fuzzy relation R = [r (i, j )]m×n and the fuzzy relation R = [min(ai , b j )]m×n [4,23]. Let us consider the OFR method based on Lop and let [b∗j ]1×n be the sequence where Lop achieves its minimum. The first step of the OFR method is to rearrange [ai ]1 × m into [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ) and rearrange the row ordering of [r(i, j)]m × n into [rˆ(i, j )]m×n according to the change of the row ordering of [ai ]1 × m . In the following, we suppose b j ∈ [aˆ p , aˆ p+1 ] with 0 ≤ p ≤ m − 1 and then obtain m + 1 candidate ( p)

values b(j0 ) , b(j1 ) , . . . , b(jm ) for b∗j . The b j and Cai [23], we know that

b(j p) =

that minimizes Lop is just the optimal value of b∗j . According to the work of Zhang

⎧ ⎪ ⎪ ⎪ aˆ p , ⎪ ⎪ ⎪ ⎨

max rˆ(i, j ) + min rˆ(i, j )

p+1≤i≤m

max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m

p+1≤i≤m

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩aˆ

2

,

p+1 ,

p+1≤i≤m

< aˆ p , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m aˆ p ≤ ≤ aˆ p+1 , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m > aˆ p+1 . 2

Actually, this result only holds under the condition that max |rˆ(i, j ) − aˆi | ≤ only related to the values of

max rˆ(i, j ) and

p+1≤i≤m

1≤i≤p

max

p+1≤i≤m

rˆ(i, j )−

min

p+1≤i≤m

2

(1)

rˆ(i, j )

( p)

. The values of b j

are not

min rˆ(i, j ) but also related to the value of max |rˆ(i, j ) − aˆi |. Considering

p+1≤i≤m

( p)

that Eq. (1) given in [23] is not correct, we supplement the computational process of b j

1≤i≤p

through considering the value of

max |rˆ(i, j ) − aˆi | in Section 3.

1≤i≤p

As the most typical representative of the existing fuzzy reasoning methods, the CRI method is very popular in fuzzy community. In [4], Cai and Zhang gave an example to illustrate the difference between the CRI method and the OFR method. The obtained results show that the OFR method is more intuitively attractive and reasonable than the CRI method. Robustness is an important feature for a fuzzy reasoning method which is concerned with how errors in fuzzy premises affect fuzzy consequences [2,3,10,11]. In Cai and Zhang’s work [4,23], the robustness of the CRI method and that of the OFR method was compared. It was shown that the OFR method can behave better in terms of robustness than the CRI method. It was pointed out in the conclusions of [23] that the analysis of the advantages and limitations of OFR methods deserved investigation. Although the OFR method is superior to the CRI method in some sense, there exist some limitations on use of OFR methods. For given fuzzy relation and fuzzy premise, a unique fuzzy consequence will be obtained by use of the CRI method. ( p) However, we will encounter some cases where there are more than one b j that can minimize objective functions. Thus ∗ there may be several different optimal values of b j , not just one. In these cases, OFR methods are not applicable. We present some examples to illustrate the limitations of OFR methods and give a further comparison between the OFR method and the CRI method in Section 3. A fuzzy implication is an important part of a fuzzy reasoning method. The fuzzy consequence of a fuzzy reasoning method is closely related to the choice of the fuzzy implication. For a given fuzzy reasoning method, we often adopt different fuzzy implications in it and analyze which one can achieve better result. In the existing OFR methods [4,23], only the Mamdani implication has been used to determine the fuzzy relation R . It is shown from the work of the robustness of fuzzy connectives that the Łukasiewicz implication, the Kleene-Dienes implication and the Zadeh implication are the most robust operators among R-implications, S-implications and QL-implications, respectively [8,10,11]. In Section 4, we respectively apply these three fuzzy implications to determine the fuzzy relation R and introduce new OFR methods based on these three fuzzy implications. The reasoning consistency of the proposed methods is also examined. In [13,14], Peng et al. proposed a fuzzy reasoning method called the pointwise optimizing fuzzy inference (POFI) method. Although they have proposed POFI solutions for several fuzzy implications, they did not present the way of calculating the POFI solutions. Section 5 discusses the relationship between the OFR method and the POFI method. According to ideas of OFR and POFI methods, we can find that POFI methods are special cases of OFR methods, where there are only one element in the universe of discourse U (i.e., let m = 1 in the fuzzy premise A = [ai ]1×m ). Therefore, determining POFI solutions of POFI methods can be converted to calculating fuzzy consequences of OFR methods.

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Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

The rest of this paper is organized as follows. In Section 2, we briefly review the framework of OFR methods. In Section 3, we give a modification of Zhang and Cai’s OFR method based on the Mamdani implication. Then we present some examples to illustrate the limitations of OFR methods and give a further comparison between the OFR method and the CRI method. In Section 4, we introduce three new OFR methods based on three robust fuzzy implications. Then we examine the reasoning consistency of the proposed methods. In Section 5, we prove that the POFI method is a special case of the OFR method. Then we obtain the POFI solutions through the computational process of OFR methods. The conclusions are examined in Section 6. 2. Preliminaries In this section, we briefly review the framework of OFR methods. Consider a set of fuzzy rules as follows: If X is A1 , then Y is B1 If X is A2 , then Y is B2 ... If X is Al , then Y is Bl

where Ai and Bi (i = 1, 2, . . . , l ) are fuzzy sets defined on universes of discourse U and V, respectively. The problem of fuzzy modus ponens (FMP) is concerned with what Y is if X is a fuzzy set A defined on U. Let μAi (x ), μBi (y ) and μA (x) be the membership functions of Ai , Bi and A, respectively. Suppose Y is B with membership function μB (y). The idea of the CRI method is that μB (y) is a composition of the given l fuzzy rules and the given fuzzy premise [20–22]. For example, consider the finite discrete universes U = {u1 , u2 , . . . , um } and V = {v1 , v2 , . . . , vn }. Let

Ri = [ri ( j1 , j2 )]m×n = [min{μAi (u j1 ), μBi (v j2 )}]m×n , R = R1 ∪ R2 ∪ . . . ∪ Rl



= [r ( j1 , j2 )]m×n = max{ri ( j1 , j2 )}



1≤i≤l

m×n

.

Then we obtain that

[μB (v j )]1×n = [μA (ui )]1×m ◦ R =



max {min{μA (ui ), r (i, j )}}

1≤i≤m

1×n

,

where min adopted in the fuzzy relation Ri is the Mamdani implication from the fuzzy premise Ai to the fuzzy consequence Bi , R composes the l fuzzy rules into a single fuzzy rule via the union operator, and thus [μB (v j )]1×n determines the fuzzy consequence B corresponding to the fuzzy premise A according to the CRI method for fuzzy modus ponens. The basic idea of the OFR method was introduced by Zhang and Cai as follows [23]: (1) Suppose a fuzzy relation (matrix) R is gained from the experience of experts (fuzzy rules) or data. Then we should trust R and treat it as a basis for evaluating the quality of a fuzzy reasoning method. (2) If a fuzzy premise A is given, then a fuzzy reasoning method will generate the corresponding consequence B. A fuzzy reasoning method is optimal if the matrix R is the closest to R in some sense, where R = A → B, denoting that A implies B. Suppose the fuzzy relation matrix is given as R = [r (i, j )]m×n and the new fuzzy premise is given as A = [ai ]1×m , which implies a new fuzzy consequence denoted as B = [b j ]1×n . Let R stand for the new fuzzy relation between A and B, which is given as R = [I (ai , b j )]m×n with I a fuzzy implication. Then the processes of the OFR method include the following two steps: (1) Choose an objective function to measure the absolute distance between R and R . The following are three examples of objective functions.

Jop =

m  n 

|r (i, j ) − I (ai , b j )|,

i=1 j=1

 Kop =

m  n 

(r (i, j ) − I (ai , b j ))2 ,

i=1 j=1

Lop =

max

1≤i≤m,1≤ j≤n

|r (i, j ) − I (ai , b j )|.

(2) Minimize the objective function to realize the optimization goal that R being closest to R.

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

655

Note that

Jop =

n 

J j , where J j =

n 

|r (i, j ) − I (ai , b j )|,

i=1

j=1 2 Kop =

m 

K j , where K j =

m 

(r (i, j ) − I (ai , b j ))2 ,

i=1

j=1

Lop = max L j , where L j = max |r (i, j ) − I (ai , b j )|. 1≤ j≤n

1≤i≤m

If the extra objective function Jj (or Kj , Lj ) achieves its minimum at b∗j ( j = 1, 2, . . . , n ), then Jop (or Kop , Lop ) achieves its minimum at B∗ = [b∗j ]1×n . In the work of Zhang and Cai [4,23], the Mamdani implication is considered as the fuzzy implication, i.e., I (ai , b j ) = IM (ai , b j ) = min(ai , b j ). Then B∗ based on Jop , Kop and Lop can be obtained in the following procedure, which includes three steps. Step 1: Rearrange A = [ai ]1×m into Aˆ = [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ). Rearrange the row ordering of R = [r (i, j )]m×n into Rˆ = [rˆ(i, j )]m×n according to the change of the row ordering of A. Step 2: Obtain b∗j ( j = 1, 2, . . . , n ) that respectively minimize Jj , Kj and Lj as follows. Let

0 = aˆ0 ≤ aˆ1 ≤ . . . ≤ aˆm ≤ aˆm+1 = 1. Suppose b j ∈ [aˆ p , aˆ p+1 ] with 0 ≤ p ≤ m − 1. Then Jj , Kj and Lj reduce to

J (j p) =

p 

|rˆ(i, j ) − aˆi | +

i=1

K j( p) =

p 

m 

(rˆ(i, j ) − aˆi )2 +

i=1

( p)

Lj

|rˆ(i, j ) − b j |,

(2)

i= p+1 m 

(rˆ(i, j ) − b j )2 ,

(3)

i= p+1

= max

 max |rˆ(i, j ) − aˆi |, max

1≤i≤p

p+1≤i≤m

|rˆ(i, j ) − b j | .

(4) ( p)

( p)

Then let us calculate values of bj that respectively minimize J j , K j

( p)

and L j

( p)

( p)

and denote them as b j , b¯ j

( p) and b˜ j .

(a) Rearrange [rˆ( p + 1, j ), rˆ( p + 2, j ), . . . , rˆ(m, j )] into [rˇ( p + 1, j ), rˇ( p + 2, j ), . . . , rˇ(m, j )] such that rˇ(i, j ) ≤ rˇ(i + 1, j ) (i = p + 1, p + 2, . . . , m − 1 ). Let

⎧ m + p m + p ˇ ˇ , j + + 1 , j r r ⎪ ⎨ 2 2 , mid = 2

⎪ ⎩rˇ m + p + 1 , j , 2

Then

 ( p)

bj

=

(b)

b¯ (j p) =

(c)

b˜ (j p) =

aˆ p , mid, aˆ p+1 ,

⎧ ⎪ ⎪ aˆ p , ⎪ ⎪ ⎨

1

m

i= p+1

rˆ(i, j ),

1 m rˆ(i, j ) < aˆ p, m − p i= p+1 1 m aˆ p ≤ rˆ(i, j ) ≤ aˆ p+1 , m − p i= p+1 1 m rˆ(i, j ) > aˆ p+1 . m − p i= p+1

⎧ ⎪ ⎪ ⎪ aˆ p , ⎪ ⎪ ⎪ ⎨

max rˆ(i, j ) + min rˆ(i, j )

p+1≤i≤m

max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m

p+1≤i≤m

2 p+1 ,

m + p is an odd number.

mid < aˆ p , aˆ p ≤ mid ≤ aˆ p+1 , mid > aˆ p+1 .

m− p ⎪ ⎪ ⎪ ⎪ ⎩aˆ p+1 ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩aˆ

m + p is an even number,

,

p+1≤i≤m

< aˆ p , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m aˆ p ≤ ≤ aˆ p+1 , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m > aˆ p+1 . 2

656

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

For p = m or b j ∈ [aˆm , 1], Jj , Kj and Lj reduce to

J (j m ) =

m 

|rˆ(i, j ) − aˆi |,

i=1

K j(m ) =

m 

(rˆ(i, j ) − aˆi )2 ,

i=1

L(jm ) = max |rˆ(i, j ) − aˆi |, 1≤i≤m

which are irrelevant to bj . Thus bj can take any value in [aˆm , 1] to minimize J (j m ) , K (j m ) and L(jm ) . Specifically, let b(jm ) , b¯ (jm ) , b˜ (jm ) = aˆm . Step 3: According to the above-mentioned procedure of calculating, it is obtained that there are m + 1 candidate values for b∗j based on any extra objective function. (a) Considering the extra objective function Jj , m + 1 candidate values b(j0 ) , b(j1 ) , . . . , b(jm ) are obtained for b∗j . Let p m   ∗ J (j p ) = |rˆ(i, j ) − aˆi | + |rˆ(i, j ) − b(j p) |. i=1

i= p+1

In this way, ∗ ∗ ∗ J ∗j = min J (j p ) , q∗ = arg min J (j p ) and b∗j = bqj .

0≤p≤m

∗ = Therefore, Jop

0≤p≤m

n

∗ j=1 J j

and the fuzzy consequence based on Jop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(b) Considering the extra objective function Kj , m + 1 candidate values b¯ (j0 ) , b¯ (j1 ) , . . . , b¯ (jm ) are obtained for b∗j . Let p m   ∗ K j( p ) = (rˆ(i, j ) − aˆi )2 + (rˆ(i, j ) − b¯ (j p) )2 . i=1

i= p+1

In this way, ∗ ∗ ∗ K ∗j = min K j( p ) , q∗ = arg min K j( p ) and b∗j = bqj .

0≤p≤m

∗ = Therefore, Kop



0≤p≤m

n j=1

K ∗j and the fuzzy consequence based on Kop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(c) Considering the extra objective function Lj , m + 1 candidate values b˜ (j0 ) , b˜ (j1 ) , . . . , b˜ (jm ) are obtained for b∗j . Let ∗ L(j p ) = max



max |rˆ(i, j ) − aˆi |, max

1≤i≤p

p+1≤i≤m

 |rˆ(i, j ) − b˜ (j p) | .

In this way, ∗ ∗ ∗ L∗j = min L(j p ) , q∗ = arg min L(j p ) and b∗j = bqj .

0≤p≤m

Therefore,

L∗op

=

0≤p≤m

max L∗ 1≤ j≤n j

and the fuzzy consequence based on Lop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

3. Some notes on OFR methods based on the Mamdani implication The first OFR method proposed by Zhang and Cai [23] is based on the Mamdani implication. In this section, we first give a modification of Zhang and Cai’s OFR method. Then we present some examples to illustrate the limitations of OFR methods and give a further comparison between the OFR method and the CRI method. ( p)

3.1. A further discussion of b j

( p) and b˜ j ( p)

( p)

The crucial problem in the calculating procedure of the OFR method is how to obtain values of bj (i.e., b j , b¯ j

˜ ( p)

( p)

( p)

( p)

and

b j ) that respectively minimize J j , K j and L j , which can be converted into finding the minimum points of the func  tions f (x ) = ni=1 |x − wi |, g(x ) = ni=1 (x − wi )2 and h(x ) = max(C, max |x − wi | ) with x ∈ [a, b](a, b ∈ [0, 1] ), wi ∈ [0, 1] and 1≤i≤n

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

C a constant. Let g (x ) = 2

x=

⎧ ⎪ b, ⎪ ⎪ ⎨

n

i=1 (x − wi )

= 0, then we have x =

1 n

n

i=1

657

wi . Hence, the value of x that minimizes g is

1 n wi , n i=1  1 n a≤ wi ≤ b, n i=1 1 n a> wi . n i=1 b<

1 n i=1 wi , ⎪ ⎪n

⎪ ⎩a,

( p) ( p) ( p) Thus b¯ j is obtained. The values of b j and b˜ j need a further discussion.

ˆ 1, w ˆ 2, . . . , w ˆ n ], where w ˆi ≤ w ˆ i+1 (i = Let us calculate the value of x that minimizes f. Rearrange [w1 , w2 , . . . , wn ] into [w 1, 2, . . . , n − 1 ). If n is an odd number, then the value of x that minimizes f(x) is

x=

⎧ ⎨b,

ˆ n+1 , w ⎩a, 2

ˆ n+1 , bw 2

If n is an even number, then the value of x that minimizes f(x) is

 x=

b < ω, a ≤ ω ≤ b, a > ω,

b, ω, a,

ˆ n,w ˆ n +1 ]. Based on the computational process of finding the value that where ω can take arbitrary value of the interval [w 2

minimizes f, we obtain the following conclusion.

2

( p)

( p)

Remark 1. The value of b j ( j = 1, 2, . . . , n; 0 ≤ p ≤ m − 1 ) that minimizes J j

 ( p)

bj

aˆ p , mid, aˆ p+1 ,

=

where

mid < aˆ p , aˆ p ≤ mid ≤ aˆ p+1 , mid > aˆ p+1 ,



rˇ1 ,

mid =

m + p + 1

rˇ

2

,j ,

m + p is an even number, m + p is an odd number,

and rˇ1 is an arbitrary value in [rˇ( m2+ p , j ), rˇ( m2+ p + 1, j )], not a single value Let us calculate the value of x that minimizes h. Let

h1 (x ) = max |x − wi | = 1≤i≤n

⎧ ⎪ ⎪ ⎨−x + max w ,

x≤

1≤i≤n

⎪ ⎪ ⎩x − min wi ,

x>

1≤i≤n

1≤i≤n

min wi + max wi

i

1≤i≤n

Note that

h1 (x ) =

⎧ ⎪ ⎪ ⎨−1 < 0,

x≤

1≤i≤n

⎪ ⎪ ⎩1 > 0,

x>

1≤i≤n

1≤i≤n

1≤i≤n

2

⎧ ⎪ ⎪ ⎪ b, ⎪ ⎪ ⎨ min w + max w i i 1≤i≤n x = 1≤i≤n , ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩

, .

min wi + max wi

b<

1≤i≤n

a≤

1≤i≤n

a>

1≤i≤n

1≤i≤n

2 min wi + max wi 1≤i≤n

2 min wi + max wi 1≤i≤n

2

1≤i≤n

2

min wi + max wi 2 min wi + max wi

1≤i≤n

2 min wi + max wi

Thus the value of x that minimizes h1 is

a,

defined as Eq. (2) can be expressed as

, ≤ b, .

, .

rˇ( m2+ p , j )+rˇ( m2+ p +1, j ) . 2

658

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

Furthermore, let h(x ) = max(C, max |x − wi | ). Then the minimal point of h is related to the value of C. If C ≤

max wi − min wi 1≤i≤n

1≤i≤n

1≤i≤n

2

,

then the value of x that minimizes h is

⎧ ⎪ ⎪ ⎪ b, ⎪ ⎪ ⎨ min w + max w i i 1≤i≤n x = 1≤i≤n , ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩

min wi + max wi

a,

If

max wi − min wi 1≤i≤n

1≤i≤n

2

x=

1≤i≤n

a≤

1≤i≤n

a>

1≤i≤n

1≤i≤n

2 min wi + max wi 1≤i≤n

2 min wi + max wi 1≤i≤n

, ≤ b, .

2

< C ≤ max wi , then the value of x that minimizes h is 1≤i≤n

⎧ ⎪ ⎨b, a,

b<

b < max wi − C , 1≤i≤n

⎪ ⎩ψ ,

a > min wi + C, 1≤i≤n

otherwise,

where ψ can take any value of the interval [max(a, max wi − C ), min(b, min wi + C )]. If C > max wi , then the value of x 1≤i≤n

1≤i≤n

1≤i≤n

that minimizes h is



x=

a,

a > min wi + C,

χ,

otherwise,

1≤i≤n

( p)

where χ can take arbitrary value of the interval [a, min(b, min wi + C )]. Since L j 1≤i≤n

= max( max |rˆ(i, j ) − aˆi |, max |rˆ(i, j ) − 1≤i≤p

p+1≤i≤m

( p) ( p) b j | ), we obtain that the value of b˜ j that minimizes L j is related to the value of max |rˆ(i, j ) − aˆi |. Based on the computa1≤i≤p

tional process of finding the value that minimizes h, we obtain the following conclusion. ( p) ( p) Remark 2. The value of b˜ j ( j = 1, 2, . . . , n; 0 ≤ p ≤ m − 1 ) that minimizes L j defined as Eq. (4) can be expressed as

(1) If max |rˆ(i, j ) − aˆi | ≤ 1≤i≤p

b˜ (j p) =

max

(2) If

p+1≤i≤m

rˆ(i, j )−

min

p+1≤i≤m

rˆ(i, j )

2

, then

⎧ ⎪ ⎪ ⎪ aˆ p , ⎪ ⎪ ⎪ ⎨

max rˆ(i, j ) + min rˆ(i, j )

< aˆ p , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m aˆ p ≤ ≤ aˆ p+1 , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m > aˆ p+1 . 2

max rˆ(i, j ) + min rˆ(i, j )

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩aˆ

p+1≤i≤m

2

,

p+1 ,

min

p+1≤i≤m

p+1≤i≤m

p+1≤i≤m

p+1≤i≤m

rˆ(i, j )−

rˆ(i, j )

2

b˜ (j p) =

max

p+1≤i≤m

⎧ ⎪ ⎨aˆ p+1 , aˆ p ,

⎪ ⎩ φ,

< max |rˆ(i, j ) − aˆi | ≤ 1≤i≤p

max rˆ(i, j ), then

p+1≤i≤m

aˆ p+1 < max rˆ(i, j ) − max |rˆ(i, j ) − aˆi |, p+1≤i≤m

1≤i≤p

aˆ p > min rˆ(i, j ) + max |rˆ(i, j ) − aˆi |, p+1≤i≤m

otherwise,

1≤i≤p

where φ can take arbitrary value of the interval [max(aˆ p , max rˆ(i, j ) − max |rˆ(i, j ) − aˆi | ), min(aˆ p+1 , min rˆ(i, j ) + p+1≤i≤m

max |rˆ(i, j ) − aˆi | )].

1≤i≤p

(3) If max |rˆ(i, j ) − aˆi | > 1≤i≤p

b˜ (j p) =



1≤i≤p

p+1≤i≤m

max rˆ(i, j ), then

p+1≤i≤m

aˆ p ,

aˆ p > min rˆ(i, j ) + max |rˆ(i, j ) − aˆi |,

σ,

otherwise,

p+1≤i≤m

1≤i≤p

where σ can take arbitrary value of the interval [aˆ p , min( min rˆ(i, j ) + max |rˆ(i, j ) − aˆi |, aˆ p+1 )]. p+1≤i≤m

1≤i≤p

According to Remark 2, we know that if we choose the objective function Lop to measure the absolute distance between R and R , then the computational process of the OFR method is quite complicated. In this case, the objective functions Jop and Kop would be better choices.

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

659

3.2. A comparison between the OFR method and the CRI method Now let us consider some examples to illustrate the difference between the OFR method and the CRI method. Considering the objective function Jop , we obtain b(j0 ) , b(j1 ) , . . . , b(jm ) for b∗j . If we plug these m + 1 candidate values into

∗ ∗ ∗ ∗ ∗ ∗ ( p∗ ) Eq. (2), then we obtain m + 1 values J (j 0 ) , J (j 1 ) , . . . , J (j m ) for Jj . Suppose J j is the minimum value of J (j 0 ) , J (j 1 ) , . . . , J (j m ) , its

( p)

corresponding b j

will be the optimized value for b∗j . Considering the objective function Kop , we obtain b¯ (j0 ) , b¯ (j1 ) , . . . , b¯ (jm )

∗ ∗ ∗ for b∗j . If we plug these m + 1 candidate values into Eq. (3), then we obtain m + 1 values K (j 0 ) , K (j 1 ) , . . . , K (j m ) for Kj . Sup-

( p∗ )

pose K j

∗ ∗ ∗ ( p) is the minimum value of K (j 0 ) , K (j 1 ) , . . . , K (j m ) , its corresponding b¯ j will be the optimized value for b∗j . In this

∗ ∗ ∗ situation, we will encounter a problem that there could be more than one minimum value among J (j 0 ) , J (j 1 ) , . . . , J (j m ) or

∗ ∗ ∗ K (j 0 ) , K (j 1 ) , . . . , K (j m ) and thus there will be more than one optimized value for b∗j . Let us consider the following two examples.





2x − y x y

Example 1. Suppose R =

and A = [x, y, z] such that x ≤ y ≤ 2x ≤ z. Then we have 0 ≤ 2x − y ≤ x ≤ y ≤ z. Considering

the OFR method based on Jop , we get b(10 ) = x, b(11 ) = ( p∗ )

2|x − y| + |y − z|. Therefore, we have J1∗ = min J1

  x y

Example 2. Suppose R =

0≤p≤3

(2 ) (3 ) x+y 2 , b1 = y and b1 ( 0∗ ) ( 1∗ ) ( 2∗ ) = J1 or J1 or J1

√ √ 2 )x+ 2y , z] 2

and A = [ (2−

∗ ∗ ∗ ∗ = z. Thus J1(0 ) = J1(1 ) = J1(2 ) = 2|x − y| and J1(3 ) =

and b∗1 = x or

=

K1

1 2 2 (x − y )

+ (y

( 0∗ )

− z )2

≥ K1

( 1∗ )

= K1

or y.

such that x ≤ y ≤ z. Then we have x ≤

Considering the OFR method based on Kop , we get b¯ (10 ) = ( 2∗ )

x+y 2

. Therefore, we

x+y ¯ ( 1 ) 2 , b1 = y and ( p∗ ) have K1∗ = min K j 0≤p≤2

x+y 2

( 0∗ )

b¯ (12 ) = z. Thus K1 ( 0∗ )

= K1

( 1∗ )

or K1

and

√ √ 2 ) x + 2y 2

≤ ( 2− ( 1∗ )

= K1 b∗1

=

1 2 (x

=

x+y 2

≤ y ≤ z.

− y )2 and

or y.

According to Examples 1 and 2, we know that if we use the OFR method to calculate the fuzzy consequence, then we will encounter some cases where more than one optimized value for b∗1 are obtained and thus we will get more than one fuzzy consequence for the same fuzzy relation and fuzzy premise. In this situation, we can consider to change another objective function. For example, if we consider the OFR method based on Kop in Example 1, then we get b¯ (10 ) = x, b¯ (11 ) = ∗ ∗ ∗ ∗ x+y ¯ ( 2 ) , b = y and b¯ (3 ) = z. Thus K (0 ) = 2(x − y )2 , K (1 ) = 3 (x − y )2 , K (2 ) = 2(x − y )2 and K (3 ) = 2(x − y )2 + (y − z )2 . Note 2

1

1

1

2

1

1

∗ ∗ ∗ ∗ ∗ ( p∗ ) that K1(1 ) ≤ K1(0 ) = K1(2 ) ≤ K1(3 ) . Therefore, we have K1∗ = min K1 = K1(1 ) and b∗1 =

(0 )

OFR method based on Jop in Example 2, then we get b1 ( 2∗ )

and J1

=

√

2 2 |x − y| +

( 0∗ )

|y − z|. Since J1

( 1∗ )

≤ J1

( 2∗ )

≤ J1

0≤p≤3 (1 ) y = x+ 2 , b1

(2 )

= y and b1

, we obtain that

J1∗

x+y 2 .

1

Meanwhile, if we consider the

∗ = z. Thus J1(0 ) =

( p∗ )

= min J1 0≤p≤2

( 0∗ )

= J1

and

( 1∗ ) 1 2 |x − y|, J1 y b∗1 = x+ 2 .

=

√

2 2 |x − y|

This is the reason why different objective functions are proposed to measure the absolute distance between R and R in the first step of the procedure of the OFR method. Although the change of the objective function can solve the problem of multiple optimized values for b∗j for a single column fuzzy relation, it may encounter some problem for a fuzzy relation with multiple columns. Let us consider the following example.



Example 3. Suppose R = √ √ ( 2 − 2 ) x + 2y 2

x y

x

√ √ ( 2 − 2 ) x + 2y 2



√ √ 2 ) x + 2y , z] 2

and A = [ (2−

such that x ≤ y ≤ z. Then we have x ≤

x+y 2

≤

≤ y ≤ z. According to Example 2, we know that if we use the OFR method based on Kop to calculate the y fuzzy consequence, then we have b∗1 = x+ or y. If we use the OFR method based on Jop to calculate the fuzzy conse2 √ √ √ √ 2 ) x + 2y , b(21 ) = (2− 22)x+ 2y 4 ∗ ( p∗ ) we have J2∗ = min J2 = J2(0 ) 0≤p≤2

quence, then we get b(20 ) = (4−

|

√

√

(2− 2 )x+ 2y 2

− z|. Therefore,

∗ ∗ and b(22 ) = z. Thus J2(0 ) = J2(1 ) =

( 1∗ )

or J2

and

b∗2

=

√

√

( 4 − 2 ) x + 2y 4

or

√

∗ 2 − y| and J2(2 ) 2 |x √ √ (2− 2 )x+ 2y . 2

=

√

2 2 |x − y|

+

According to Example 3, we know that the OFR method does not work regardless of what objective function is chosen (Jop or Kop ). The CRI method can confirm that only one fuzzy consequence is obtained for given fuzzy relation and fuzzy premise. √

√

In Examples 1–3, if we use the CRI method, then we obtain that B∗ = [y], B∗ = [max(x, min(y, z ))] and B∗ = [y, (2− 22)x+ 2y ]. In this sense, the CRI method could be more effective than the OFR method. However, the OFR method will be more sensitive than the CRI method in some case. Let us consider the following example.

 

x and A = [a1 , a2 ] such that x ≤ y and a1 ≤ a2 . Table 1 lists the reasoning results for the CRI and y OFR methods. The CRI method generates a uniform result of B∗ = [y] when x ≤ y ≤ a2 irrespective of the value of a1 . For the y ∗ OFR method, a1 ≤ x ≤ y ≤ a2 implies B∗ = [y], x ≤ a1 ≤ y ≤ a2 implies B∗ = [y] or [ x+ 2 ] and x ≤ y ≤ a1 ≤ a2 implies B = [a1 ] or y ∗ changes as the change of a . For x ≤ a ≤ y ≤ a , x ≤ y ≤ a ≤ a and x ≤ a ≤ a ≤ y, the values of B∗ [ x+ ] . That is to say, B 1 1 2 1 2 1 2 2 have two choices. They depend on the values of a1 and a2 . In this sense, the OFR can better reflect the influence of the given Example 4. Suppose R =

660

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669 Table 1 The reasoning results for the CRI and OFR methods. Method

a1 ≤ x ≤ y ≤ a2

x ≤ a1 ≤ y ≤ a2

x ≤ y ≤ a1 ≤ a2

CRI OFR

[y] [y]

[y] [ x+2 y ] or [y]

[y] [ x+2 y ] or [a1 ]

Method CRI OFR

a1 ≤ a2 ≤ x ≤ y [a2 ] [a2 ]

a1 ≤ x ≤ a2 ≤ y [a2 ] [a2 ]

x ≤ a1 ≤ a2 ≤ y [a2 ] [ x+2 y ] or [a2 ]

premise to the fuzzy consequence. For a1 ≤ a2 ≤ x ≤ y and a1 ≤ x ≤ a2 ≤ y, the reasoning results for the CRI and OFR methods are the same because these two methods are equivalent under these two conditions. It is shown from the following propositions that although the basic idea of the OFR method is different from that of the CRI method, these two methods can be equivalent under certain conditions. Proposition 1. Let the fuzzy premise A = [ai ]1×m and the fuzzy relation R = [r (i, j )]m×n . If ai ≤ min r (i, j )(i = 1, 2, . . . , m ), then 1≤ j≤n

the fuzzy consequence B = [b j ]1×n determined by the CRI method is equal to B∗ = [b∗j ]1×n determined by OFR methods based on Jop , Kop and Lop . Proof. Rearrange A = [ai ]1×m into Aˆ = [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ) and rearrange the respective row ordering of R = [r (i, j )]m×n into Rˆ = [rˆ(i, j )]m×n . If B is determined by the CRI method, then we obtain that B = [b j ]1×n = [ max {min(ai , r (i, j ))}]1×n = [ max ai ]1×n = 1≤i≤m

1≤i≤m

[aˆm ]1×n . Consider the fuzzy consequence determined by the OFR method based on Jop . Rearrange [rˆ( p + 1, j ), rˆ( p + 2, j ), . . . , rˆ(m, j )] into [rˇ( p + 1, j ), rˇ( p + 2, j ), . . . , rˇ(m, j )] such that rˇ(i, j ) ≤ rˇ(i + 1, j ) (i = p + 1, p + 2, . . . , m − 1 ). Since min(rˆ( p + 1, j ), rˆ( p + 2, j ), . . . , rˆ(m, j )) ≥ min(aˆ p+1 , aˆ p+2 , . . . , aˆm ) = aˆ p+1 ( p = 0, 1, . . . , m − 1 ), we have min(rˇ( p + 1, j ), rˇ( p + 2, j ), . . . , rˇ(m, j )) ≥ aˆ p+1 . If m + p is an odd number, then rˇ( m+2p+1 , j ) ≥ aˆ p+1 . If m + p is an even number, then rˇ( m2+ p , j )+rˇ( m2+ p +1, j ) 2

( p) ≥ aˆ p+1 . Therefore, we have b j = aˆ p+1 ( p = 0, 1, . . . , m − 1 ) and b(jm−1 ) = aˆm = b(jm ) . For 0 ≤ p ≤ m − p m  p−1  ( p∗ ) (( p−1 )∗ ) ˆ p |. = i=1 |rˆ(i, j ) − aˆi | + i= p+1 |rˆ(i, j ) − aˆ p+1 | and J j = i=1 |rˆ(i, j ) − aˆi | + m 1, it is obtained that J j i= p |rˆ(i, j ) − a ∗ ∗   (( p−1 ) ) (p ) m m Therefore, we have J j − Jj = i= p+1 |rˆ(i, j ) − aˆ p | − i= p+1 |rˆ(i, j ) − aˆ p+1 |. Since rˆ(i, j ) − aˆ p ≥ rˆ(i, j ) − aˆ p+1 ≥ 0 (i = ∗  m p−1 )∗ ) ( p∗ ) ˆ p| ≥ m ˆ p+1 | and thus J (( p + 1, p + 2, . . . , m ), we have ≥ J j . Then we have J (j 0 ) ≥ i= p+1 |rˆ(i, j ) − a i= p+1 |rˆ(i, j ) − a j ∗ ∗ ∗ ( p∗ ) m−1 )∗ ) m−1 )∗ ) = J (j m ) . Hence, J ∗j = min J j = J (( = J (j m ) and b∗j = b(jm−1 ) = b(jm ) = aˆm . Therefore, it is obtained J (j 1 ) ≥ · · · ≥ J (( j j

0≤p≤m

that B∗ = [b∗j ]1×n = [aˆm ]1×n . Consider the fuzzy consequence determined by the OFR method based on Kop . If 0 ≤ p ≤ m − 1, then m  1 1 rˆ(i, j ) = (rˆ( p + 1, j ) + rˆ( p + 2, j ) + · · · + rˆ(m, j )) m− p m− p i= p+1

1 (aˆ p+1 + aˆ p+2 + . . . + aˆm ) m− p 1 ≥ (aˆ p+1 + aˆ p+1 + · · · + aˆ p+1 ) m− p    ≥

m−p

= aˆ p+1 . ( p) Therefore, we have b¯ j = aˆ p+1 ( p = 0, 1, . . . , m − 1 ) and b¯ (jm−1 ) = aˆm = b¯ (jm ) . For 0 ≤ p ≤ m − 1, it is obtained that p   p−1  ( p∗ ) p−1 )∗ ) ˆ p+1 )2 and K (( ˆ p )2 . Hence, we obKj = i=1 (rˆ(i, j ) − aˆi )2 + m = i=1 (rˆ(i, j ) − aˆi )2 + m i= p+1 (rˆ(i, j ) − a i= p (rˆ(i, j ) − a j ∗ ∗   (( p−1 ) ) (p ) m m 2 2 tain that K j − Kj = i= p+1 (rˆ(i, j ) − aˆ p ) − i= p+1 (rˆ(i, j ) − aˆ p+1 ) . Since rˆ(i, j ) − aˆ p ≥ rˆ(i, j ) − aˆ p+1 ≥ 0 (i = p + ∗ m  p−1 )∗ ) ( p∗ ) ˆ p )2 ≥ m ˆ p+1 )2 and thus K (( 1, p + 2, . . . , m ), we have ≥ K j . Then we have K (j 0 ) ≥ i= p+1 (rˆ(i, j ) − a i= p+1 (rˆ(i, j ) − a j ∗ ∗ ∗ ( p∗ ) m−1 )∗ ) m−1 )∗ ) = K (j m ) . Hence, K ∗j = min K j = K (( = K (j m ) and b∗j = b¯ (jm−1 ) = b¯ (jm ) = aˆm . Therefore, it is obK (j 1 ) ≥ . . . ≥ K (( j j

0≤p≤m

tained that B∗ = [b∗j ]1×n = [aˆm ]1×n . Consider the fuzzy consequence determined by the OFR method based on Lop . Note that min(rˆ( p + 1, j ), rˆ( p + 2, j ), . . . , rˆ(m, j )) ≥ min(aˆ p+1 , aˆ p+2 , . . . , aˆm ) = aˆ p+1 ( p = 0, 1, . . . , m − 1 ). If max |rˆ(i, j ) − aˆi | ≤ max

we obtain that

p+1≤i≤m

rˆ(i, j )+ 2

min

p+1≤i≤m

rˆ(i, j )

( p) ≥ aˆ p+1 and b˜ j = aˆ p+1 . If

max

p+1≤i≤m

1≤i≤p rˆ(i, j )− min

p+1≤i≤m

2

rˆ(i, j )

max

p+1≤i≤m

rˆ(i, j )−

min

p+1≤i≤m

2

< max |rˆ(i, j ) − aˆi | ≤ 1≤i≤p

rˆ(i, j )

, then

max rˆ(i, j ),

p+1≤i≤m

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

then

considering

min rˆ(i, j ) + max |rˆ(i, j ) − aˆi | > aˆ p+1 ≥ aˆ p

that

p+1≤i≤m

1≤i≤p

we

have

661

( p) b˜ j = aˆ p+1

or

φ . Note that φ =

[max(aˆ p , max rˆ(i, j ) − max |rˆ(i, j ) − aˆi | ), aˆ p+1 ] and thus φ can take the value of aˆ p+1 . If max |rˆ(i, j ) − aˆi | > p+1≤i≤m

1≤i≤p

then considering that

1≤i≤p

max rˆ(i, j ),

p+1≤i≤m

( p) min rˆ(i, j ) + max |rˆ(i, j ) − aˆi | > aˆ p+1 ≥ aˆ p we have b˜ j = σ . Note that σ = [aˆ p , min( min rˆ(i, j ) +

p+1≤i≤m

p+1≤i≤m

1≤i≤p

( p) max |rˆ(i, j ) − aˆi |, aˆ p+1 )] = [aˆ p , aˆ p+1 ] and thus σ can take the value of aˆ p+1 . Hence, we have b˜ j = aˆ p+1 ( p = 0, 1, . . . , m − 1 )

1≤i≤p

( p∗ ) (( p−1 )∗ ) and b˜ (jm−1 ) = aˆm = b˜ (jm ) . For 0 ≤ p ≤ m − 1, we have L j = max( max |rˆ(i, j ) − aˆi |, max |rˆ(i, j ) − aˆ p+1 | ) and L j = ( p∗ )

max( max |rˆ(i, j ) − aˆi |, max |rˆ(i, j ) − aˆ p | ) ≥ L j 1≤i≤p

( p∗ )

L∗j = min L j 0≤p≤m

p+1≤i≤m

1≤i≤p

p+1≤i≤m

∗ ∗ ∗ m−1 )∗ ) . Then we obtain that L(j0 ) ≥ L(j1 ) ≥ · · · ≥ L(( = L(jm ) . Hence, j

∗ m−1 )∗ ) = L(( = L(jm ) and b∗j = b˜ (jm−1 ) = b˜ (jm ) = aˆm . Therefore, it is obtained that B∗ = [b∗j ]1×n = [aˆm ]1×n . j

Since B = B∗ = [aˆm ]1×n , we conclude that the fuzzy consequence B determined by the CRI method is equal to B∗ determined by the OFR methods.  Proposition 2. Let the fuzzy premise A = [ai ]1×m and the fuzzy relation R = [r (i, j )]m×n . If m = 1, then the fuzzy consequence B = [b j ]1×n determined by the CRI method is equal to B∗ = [b∗j ]1×n determined by the OFR methods based on Jop , Kop and Lop . Proof. If m = 1, then the fuzzy consequence determined by the CRI method and the fuzzy consequences determined by the OFR methods based on Jop , Kop and Lop are all equal to [min (a1 , r(i, j))]1 × n .  Remark 3. It is shown that the OFR methods based on Jop , Kop and Lop are equivalent under the conditions of Propositions 1 and 2. 4. New OFR methods based on three robust fuzzy implications In this section, we introduce new OFR methods based on three robust fuzzy implications. Then we examine the reasoning consistency of the proposed methods. It is shown from the work of the robustness of fuzzy connectives that the Łukasiewicz implication IL (x, y ) = min(1 − x + y, 1 ), the Kleene-Dienes implication IK (x, y ) = max(1 − x, y ) and the Zadeh implication IZ (x, y ) = max(1 − x, min(x, y )) are the most robust operators among R-implications, S-implications and QL-implications, respectively [8,10,11]. In the following, we apply IL , IK and IZ to the OFR method and study how the fuzzy consequence [b∗j ]1×n corresponding to the fuzzy premise [ai ]1 × m is determined through minimizing the given objective functions. 4.1. The OFR method based on the Łukasiewicz implication The algorithm for the OFR method based on IL can be presented as follows. Step 1: Rearrange A = [ai ]1×m into Aˆ = [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ). Rearrange the row ordering of R = [r (i, j )]m×n into Rˆ = [rˆ(i, j )]m×n according to the change of the row ordering of A. Step 2: Obtain b∗j ( j = 1, 2, . . . , n ) that minimize Jj , Kj and Lj as follows. Let

0 = aˆ0 ≤ aˆ1 ≤ · · · ≤ aˆm ≤ aˆm+1 = 1. Suppose b j ∈ [aˆ p , aˆ p+1 ] with 0 ≤ p ≤ m − 1. Then Jj , Kj and Lj reduce to

J (j p) =

p  i=1

K j( p) =

p 

Lj

|rˆ(i, j ) − 1 + aˆi − b j |,

i= p+1

(rˆ(i, j ) − 1 )2 +

i=1

( p)

m 

|rˆ(i, j ) − 1| +

(rˆ(i, j ) − 1 + aˆi − b j )2 ,

i= p+1



= max

m 



max |rˆ(i, j ) − 1|, max

1≤i≤p

p+1≤i≤m

|rˆ(i, j ) − 1 + aˆi − b j | . ( p)

( p)

Then let us calculate values of bj that respectively minimize J j , K j

( p)

and L j

( p)

( p)

and denote them as b j , b¯ j

( p) and b˜ j .

(a) Let rˆ(i, j ) − 1 + aˆi = zi (i = p + 1, p + 2, . . . , m ), then [rˆ( p + 1, j ) − 1 + aˆ p+1 , rˆ( p + 2, j ) − 1 + aˆ p+2 , . . . , rˆ(m, j ) − 1 + aˆm ] = [z p+1 , z p+2 , . . . , zm ]. Rearrange [z p+1 , z p+2 , . . . , zm ] into [zˆ p+1 , zˆ p+2 , . . . , zˆm ] such that zˆi ≤ zˆi+1 (i = p + 1, p + 2, . . . , m − 1 ). Let



mid =

rˇ1 ,

m + p + 1

zˆ

2

,j ,

m + p is an even number, m + p is an odd number,

662

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

where rˇ1 is an arbitrary value in [zˆ( m2+ p , j ), zˆ( m2+ p + 1, j )]. Then



( p)

bj

(b) Let mid =

aˆ p , mid, aˆ p+1 ,

=

m

1 m−p

i= p+1 (rˆ(i,



aˆ p , mid, aˆ p+1 ,

b¯ (j p) =

(c) Let mid =

mid < aˆ p , aˆ p ≤ mid ≤ aˆ p+1 , mid > aˆ p+1 .

mid < aˆ p , aˆ p ≤ mid ≤ aˆ p+1 , mid > aˆ p+1 .

max (rˆ(i, j )−1+aˆi )+

p+1≤i≤m

min

p+1≤i≤m

(rˆ(i, j )−1+aˆi )

2



aˆ p , mid, aˆ p+1 ,

b˜ (j p) =

If

j ) + aˆi − 1 ). Then

max (rˆ(i, j )−1+aˆi )−

p+1≤i≤m

bj

=

φ

where

min

p+1≤i≤m

(rˆ(i, j )−1+aˆi )

< max |rˆ(i, j ) − 1| ≤

, then

p+1≤i≤m

aˆ p+1 < max (rˆ(i, j ) − 1 + aˆi ) − max |rˆ(i, j ) − 1|, p+1≤i≤m

1≤i≤p

aˆ p > min (rˆ(i, j ) − 1 + aˆi ) + max |rˆ(i, j ) − 1|,

aˆ p ,

⎪ ⎩ φ,

p+1≤i≤m

1≤i≤p

otherwise, take

arbitrary

p+1≤i≤m



(rˆ(i, j )−1+aˆi )

max (rˆ(i, j ) − 1 + aˆi ), then

1≤i≤p

value

of

the

interval

[max(aˆ p , max (rˆ(i, j ) − 1 + aˆi ) − max |rˆ(i, j ) − p+1≤i≤m

1| ), min(aˆ p+1 , min (rˆ(i, j ) − 1 + aˆi ) + max |rˆ(i, j ) − 1| )]. If max |rˆ(i, j ) − 1| >

b˜ (j p) =

min

p+1≤i≤m

2

mid < aˆ p , aˆ p ≤ mid ≤ aˆ p+1 , mid > aˆ p+1 .

⎧ ⎪ ⎨aˆ p+1 ,

can

max (rˆ(i, j )−1+aˆi )−

p+1≤i≤m

1≤i≤p

2

˜ ( p)

. If max |rˆ(i, j ) − 1| ≤

1≤i≤p

1≤i≤p

aˆ p ,

aˆ p > min (rˆ(i, j ) − 1 + aˆi ) + max |rˆ(i, j ) − 1|,

σ,

otherwise,

p+1≤i≤m

1≤i≤p

max (rˆ(i, j ) − 1 + aˆi ), then

p+1≤i≤m

1≤i≤p

where σ can take any value of the interval [aˆ p , min( min (rˆ(i, j ) − 1 + aˆi ) + max |rˆ(i, j ) − 1|, aˆ p+1 )]. p+1≤i≤m

1≤i≤p

For p = m or b j ∈ [aˆm , 1], Jj , Kj and Lj reduce to

J (j m ) =

m 

|rˆ(i, j ) − 1|,

i=1

K j(m ) =

m 

(rˆ(i, j ) − 1 )2 ,

i=1

L(jm ) = max |rˆ(i, j ) − 1|, 1≤i≤m

which are irrelevant to bj . Thus bj can take any value in [aˆm , 1] to minimize J (j m ) , K (j m ) and L(jm ) . Specifically, let b(jm ) , b¯ (jm ) , b˜ (jm ) = aˆm . Step 3: According to the above-mentioned procedure of calculating, it is obtained that there are m + 1 candidate values for b∗j based on any extra objective function. (a) Considering the extra objective function Jj , m + 1 candidate values b(j0 ) , b(j1 ) , . . . , b(jm ) are obtained for b∗j . Let p m   ∗ J (j p ) = |rˆ(i, j ) − 1| + |rˆ(i, j ) − 1 + aˆi − b(j p) |. i=1

i= p+1

In this way, ∗ ∗ ∗ J ∗j = min J (j p ) , q∗ = arg min J (j p ) and b∗j = bqj .

0≤p≤m

Therefore,

∗ Jop

=

n

∗ j=1 J j

0≤p≤m

and the fuzzy consequence based on Jop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(b) Considering the extra objective function Kj , m + 1 candidate values b¯ (j0 ) , b¯ (j1 ) , . . . , b¯ (jm ) are obtained for b∗j . Let p m   ∗ K j( p ) = (rˆ(i, j ) − 1 )2 + (rˆ(i, j ) − 1 + aˆi − b¯ (j p) )2 . i=1

i= p+1

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

663

In this way, ∗ ∗ ∗ K ∗j = min K j( p ) , q∗ = arg min K j( p ) and b∗j = bqj .

0≤p≤m



∗ = Therefore, Kop

0≤p≤m

n j=1

K ∗j and the fuzzy consequence based on Kop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(c) Considering the extra objective function Lj , m + 1 candidate values b˜ (j0 ) , b˜ (j1 ) , . . . , b˜ (jm ) are obtained for b∗j . Let ( p∗ )

Lj



= max



max |rˆ(i, j ) − 1|, max |rˆ(i, j ) − 1 + aˆi − b˜ (j p) | .

1≤i≤p

p+1≤i≤m

In this way, ∗ ∗ ∗ L∗j = min L(j p ) , q∗ = arg min L(j p ) and b∗j = bqj .

0≤p≤m

Therefore,

L∗op

0≤p≤m

max L∗ 1≤ j≤n j

=

and the fuzzy consequence based on Lop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

4.2. The OFR method based on the Kleene-Dienes implication The algorithm for the OFR method based on IK can be presented as follows. Step 1: Rearrange A = [ai ]1×m into Aˆ = [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ). Rearrange the row ordering of R = [r (i, j )]m×n into Rˆ = [rˆ(i, j )]m×n according to the change of the row ordering of A. Step 2: Obtain b∗j ( j = 1, 2, . . . , n ) that minimize Jj , Kj and Lj as follows. Let

0 = aˆ0 ≤ aˆ1 ≤ · · · ≤ aˆm ≤ aˆm+1 = 1. Then we have

0 = 1 − aˆm+1 ≤ 1 − aˆm ≤ · · · ≤ 1 − aˆ1 ≤ 1 − aˆ0 = 1. Suppose b j ∈ [1 − aˆ p+1 , 1 − aˆ p ] with 0 ≤ p ≤ m − 1. Then Jj , Kj and Lj reduce to

J (j p) =

p  i=1

K j( p) =

p 

Lj

|rˆ(i, j ) − b j |.

i= p+1

(rˆ(i, j ) − 1 + aˆi )2 +

i=1

( p)

m 

|rˆ(i, j ) − 1 + aˆi | +

(rˆ(i, j ) − b j )2 .

i= p+1



= max

m 



max |rˆ(i, j ) − 1 + aˆi |, max

1≤i≤p

p+1≤i≤m

|rˆ(i, j ) − b j | . ( p)

( p)

Then let us calculate values of bj that respectively minimize J j , K j

( p)

and L j

( p)

( p)

and denote them as b j , b¯ j

( p) and b˜ j .

(a) Rearrange [rˆ( p + 1, j ), rˆ( p + 2, j ), . . . , rˆ(m, j )] into [rˇ( p + 1, j ), rˇ( p + 2, j ), . . . , rˇ(m, j )] such that rˇ(i, j ) ≤ rˇ(i + 1, j ) (i = p + 1, p + 2, . . . , m − 1 ). Let



mid =

rˇ2 ,

m + p + 1

rˇ

2

,j ,

m + p is an even number, m + p is an odd number.

where rˇ2 is an arbitrary value in [rˇ( m2+ p , j ), rˇ( m2+ p + 1, j )]. Then we have



( p)

bj

=

(b)

b¯ (j p) =

1 − aˆ p+1 , mid, 1 − aˆ p ,

⎧ ⎪ ⎪ 1 − aˆ p+1 , ⎪ ⎪ ⎨  1

m− p ⎪ ⎪ ⎪ ⎪ ⎩1 − aˆ p ,

mid < 1 − aˆ p+1 , 1 − aˆ p+1 ≤ mid ≤ 1 − aˆ p, mid > 1 − aˆ p.

m i= p+1

rˆ(i, j ),

1 m rˆ(i, j ) < 1 − aˆ p+1 , m − p i= p+1 1 m 1 − aˆ p+1 ≤ rˆ(i, j ) ≤ 1 − aˆ p, m − p i= p+1 1 m rˆ(i, j ) > 1 − aˆ p. m − p i= p+1

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Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

(c) If max |rˆ(i, j ) − 1 + aˆi | ≤

max

p+1≤i≤m

rˆ(i, j )−

min

p+1≤i≤m

rˆ(i, j )

2

1≤i≤p

, then

⎧ ⎪ ⎪ ⎪ 1 − aˆ p+1 , ⎪ ⎪ ⎪ ⎨ max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m b˜ (j p) = p+1≤i≤m , ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩1 − aˆ p ,

max

If

p+1≤i≤m

rˆ(i, j )−

min

p+1≤i≤m

rˆ(i, j )

˜ ( p)

bj

=

1≤i≤p

max rˆ(i, j ), then

p+1≤i≤m

1 − aˆ p < max rˆ(i, j ) − max |rˆ(i, j ) − 1 + aˆi |,

1 − aˆ p+1 ,

⎪ ⎩ φ,

p+1≤i≤m

< 1 − aˆ p+1 , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m 1 − aˆ p+1 ≤ ≤ 1 − aˆ p , 2 max rˆ(i, j ) + min rˆ(i, j ) p+1≤i≤m p+1≤i≤m > 1 − aˆ p . 2

< max |rˆ(i, j ) − 1 + aˆi | ≤

2

⎧ ⎪ ⎨1 − aˆ p ,

max rˆ(i, j ) + min rˆ(i, j )

p+1≤i≤m

p+1≤i≤m

1≤i≤p

1 − aˆ p+1 > min rˆ(i, j ) + max |rˆ(i, j ) − 1 + aˆi |, otherwise,

p+1≤i≤m

1≤i≤p

φ can take any value of the interval [max(1 − aˆ p+1 , max rˆ(i, j ) − max |rˆ(i, j ) − 1 + aˆi | ), min(1 −

where

aˆ p , min rˆ(i, j ) + max |rˆ(i, j ) − 1 + aˆi | )]. If max |rˆ(i, j ) − 1 + aˆi | > p+1≤i≤m

b˜ (j p) =



1≤i≤p

1≤i≤p

p+1≤i≤m

max rˆ(i, j ), then

p+1≤i≤m

1 − aˆ p+1 ,

1 − aˆ p+1 > min rˆ(i, j ) + max |rˆ(i, j ) − 1 + aˆi |,

σ,

otherwise,

p+1≤i≤m

1≤i≤p

1≤i≤p

where σ can take any value of the interval [1 − aˆ p+1 , min( min rˆ(i, j ) + max |rˆ(i, j ) − 1 + aˆi |, 1 − aˆ p )]. p+1≤i≤m

1≤i≤p

For p = m or b j ∈ [0, 1 − aˆm ], Jj , Kj and Lj reduce to

J (j m ) =

m 

|rˆ(i, j ) − 1 + aˆi |,

i=1

K j(m ) =

m 

(rˆ(i, j ) − 1 + aˆi )2 ,

i=1

L(jm ) = max |rˆ(i, j ) − 1 + aˆi |, 1≤i≤m

which are irrelevant to bj . Thus bj can take any value in [0, 1 − aˆm ] to minimize J (j m ) , K (j m ) and L(jm ) . Specifically, let b(jm ) , b¯ (jm ) , b˜ (jm ) = 1 − aˆm . Step 3: According to the above-mentioned procedure of calculating, it is obtained that there are m + 1 candidate values for b∗j based on any extra objective function. (a) Considering the extra objective function Jj , m + 1 candidate values b(j0 ) , b(j1 ) , . . . , b(jm ) are obtained for b∗j . Let p m   ∗ J (j p ) = |rˆ(i, j ) − 1 + aˆi | + |rˆ(i, j ) − b(j p) |. i=1

i= p+1

In this way, ∗ ∗ ∗ J ∗j = min J (j p ) , q∗ = arg min J (j p ) and b∗j = bqj .

0≤p≤m

Therefore,

∗ Jop

=

0≤p≤m

n

∗ j=1 J j

and the fuzzy consequence based on Jop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(b) Considering the extra objective function Kj , m + 1 candidate values b¯ (j0 ) , b¯ (j1 ) , . . . , b¯ (jm ) are obtained for b∗j . Let p m   ∗ K j( p ) = (rˆ(i, j ) − 1 + aˆi )2 + (rˆ(i, j ) − b¯ (j p) )2 . i=1

i= p+1

In this way, ∗ ∗ ∗ K ∗j = min K j( p ) , q∗ = arg min K j( p ) and b∗j = bqj .

0≤p≤m

∗ = Therefore, Kop



0≤p≤m

n j=1

K ∗j and the fuzzy consequence based on Kop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

665

(c) Considering the extra objective function Lj , m + 1 candidate values b˜ (j0 ) , b˜ (j1 ) , . . . , b˜ (jm ) are obtained for b∗j . Let ∗ L(j p ) = max



max |rˆ(i, j ) − 1 + aˆi |, max

1≤i≤p

p+1≤i≤m

 |rˆ(i, j ) − b˜ (j p) | .

In this way, ∗ ∗ ∗ L∗j = min L(j p ) , q∗ = arg min L(j p ) and b∗j = bqj .

0≤p≤m

0≤p≤m

Therefore, L∗op = max L∗j and the fuzzy consequence based on Lop is B∗ = [b∗1 , b∗2 , . . . , b∗n ]. 1≤ j≤n

4.3. The OFR method based on the Zadeh implication The algorithm for the OFR method based on IZ can be presented as follows. Step 1: Rearrange A = [ai ]1×m into Aˆ = [aˆi ]1×m such that aˆi ≤ aˆi+1 (i = 1, 2, . . . , m − 1 ). Rearrange the row ordering of R = [r (i, j )]m×n into Rˆ = [rˆ(i, j )]m×n according to the change of the row ordering of A. Step 2: Obtain b∗j ( j = 1, 2, . . . , n ) that minimize Jj , Kj and Lj as follows. Let

0 = aˆ0 ≤ aˆ1 ≤ · · · ≤ aˆm ≤ aˆm+1 = 1. Suppose h ∈ {0, 1, . . . , m − 1} such that aˆh ≤ reduce to

J (j p) =

h  i=1

K j( p) = Lj

≤ aˆh+1 and b j ∈ [aˆh+ p , aˆh+ p+1 ] with 0 ≤ p ≤ m − h − 1. Then Jj , Kj and Lj

|rˆ(i, j ) − aˆi | +

i=h+1

h 

(rˆ(i, j ) − 1 + aˆi )2 +

i=1

( p)

h+ p 

|rˆ(i, j ) − 1 + aˆi | +

1 2

|rˆ(i, j ) − b j |,

i=h+ p+1

h+ p 

m 

(rˆ(i, j ) − aˆi )2 +

i=h+1



max |rˆ(i, j ) − 1 + aˆi |,

= max

m 

1≤i≤h

(rˆ(i, j ) − b j )2 ,

i=h+ p+1

max

h+1≤i≤h+ p

|rˆ(i, j ) − aˆi |,

 max

h+ p+1≤i≤m

( p)

( p)

Then let us calculate values of bj that respectively minimize J j , K j

|rˆ(i, j ) − b j | . ( p)

( p)

( p)

and denote them as b j , b¯ j

and L j

( p) and b˜ j .

(a) Rearrange [rˆ(h + p + 1, j ), rˆ(h + p + 2, j ), . . . , rˆ(m, j )] into [rˇ(h + p + 1, j ), rˇ(h + p + 2, j ), . . . , rˇ(m, j )] such that rˇ(i, j ) ≤ rˇ(i + 1, j ) (i = h + p + 1, h + p + 2, . . . , m − 1 ). Let

mid =

⎧ ⎨rˇ3 ,



m+h+ p+1 ,j , ⎩rˇ 2

m + h + p is an even number, m + h + p is an odd number.

where rˇ3 is an arbitrary value in [rˇ( m+2h+ p , j ), rˇ( m+2h+ p + 1, j )]. Then we have



( p)

bj (b)

=

aˆh+ p , mid, aˆh+ p+1 ,

mid < aˆh+ p, aˆh+ p ≤ mid ≤ aˆh+ p+1 , mid > aˆh+ p+1 .

⎧ ⎨aˆh+ p,  m 1 ¯b( p) = rˆ(i, j ), j ⎩amˆ −h−p , i=h+ p+1 h+ p+1

max

(c) Let aˆi |,

h+ p+1≤i≤m

min

h+ p+1≤i≤m

rˆ(i, j )

2

max

h+1≤i≤h+ p

max



h+ p+1≤i≤m

(

)

i=h+ p+1 rˆ i, j <  aˆh+ p ≤ m−h1 −p m i=h+ p+1 rˆ  m 1 i=h+ p+1 rˆ i, j > m−h−p

(

max

h+ p+1≤i≤m

and

aˆh+ p, (i, j ) ≤ aˆh+ p+1 , aˆh+ p+1 .

)

rˆ(i, j )+

min

h+ p+1≤i≤m

rˆ(i, j )

2

aˆh+ p , mid, aˆh+ p+1 ,

= mid.

If

1≤i≤h

min

h+ p+1≤i≤m



aˆh+ p+1 , aˆh+ p , φ,

max( max |rˆ(i, j ) − 1 + 1≤i≤h

mid < aˆh+ p, aˆh+ p ≤ mid ≤ aˆh+ p+1 , mid > aˆh+ p+1 .

rˆ(i, j ) − max( max |rˆ(i, j ) − 1 + aˆi |,

|rˆ(i, j ) − aˆi | ) + b˜ (j p) =

=

m

|rˆ(i, j ) − aˆi | ) ≤ , then

b˜ (j p) = Let

rˆ(i, j )−

1 m−h−p

max

h+1≤i≤h+ p

|rˆ(i, j ) − aˆi | ) =  and max( max |rˆ(i, j ) − 1 + aˆi |,

rˆ(i, j ) = . If  < max( max |rˆ(i, j ) − 1 + aˆi |, 1≤i≤h

aˆh+ p+1 < , aˆh+ p > , otherwise,

1≤i≤h

max

h+1≤i≤h+ p

|rˆ(i, j ) − aˆi | ) ≤

max

h+ p+1≤i≤m

max

h+1≤i≤h+ p

rˆ(i, j ), then

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Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

φ can take any value of the interval [max(aˆh+ p, ), min(aˆh+ p+1 , )]. If max( max |rˆ(i, j ) − 1 +

where aˆi |,

max

h+1≤i≤h+ p

|rˆ(i, j ) − aˆi | ) >



b˜ (j p) =

max

h+ p+1≤i≤m

1≤i≤h

rˆ(i, j ), then

aˆh+ p > , otherwise,

aˆh+ p , σ,

where σ can take any value of the interval [aˆh+ p , min( , aˆh+ p+1 )]. For p = m − h or b j ∈ [aˆm , 1], Jj , Kj and Lj reduce to

J (j m−h ) =

h  i=1

K j(m−h ) = Lj

|rˆ(i, j ) − aˆi |,

i=h+1

h  i=1

(m−h )

m 

|rˆ(i, j ) − 1 + aˆi | + (rˆ(i, j ) − 1 + aˆi )2 +

m 

(rˆ(i, j ) − aˆi )2 ,

i=h+1



= max max |rˆ(i, j ) − 1 + aˆi |, max 1≤i≤h

h+1≤i≤m

|rˆ(i, j ) − aˆi | ,

which are irrelevant to bj . Thus bj can take any value in [aˆm , 1] to minimize J (j m−h ) , K (j m−h ) and L(jm−h ) . Specifically, let b(jm−h ) , b¯ (jm−h ) , b˜ (jm−h ) = aˆm . If h = m, then Jj , Kj and Lj reduce to

Jj =

m 

|rˆ(i, j ) − 1 + aˆi |,

i=1

Kj =

m 

(rˆ(i, j ) − 1 + aˆi )2 ,

i=1

L j = max |rˆ(i, j ) − 1 + aˆi |, 1≤i≤m

which are irrelevant to bj . In this sense, bj can take any value in [0, 1] to minimize Jj , Kj and Lj . Step 3: According to the above-mentioned procedure of calculating, if h = m, then there are m − h + 1 candidate values for b∗j based on any extra objective function. (a) Considering the extra objective function Jj , m − h + 1 candidate values b(j0 ) , b(j1 ) , . . . , b(jm−h ) are obtained for b∗j . Let h+ p h   ∗ J (j p ) = |rˆ(i, j ) − 1 + aˆi | + |rˆ(i, j ) − aˆi | + i=1

i=h+1

m 

|rˆ(i, j ) − b(j p) |.

i=h+ p+1

In this way, ∗ ∗ ∗ min J (j p ) , q∗ = arg min J (j p ) and b∗j = bqj .

J ∗j = Therefore,

0≤p≤m−h

∗ Jop

=

0≤p≤m−h

n

∗ j=1 J j

and the fuzzy consequence based on Jop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(b) Considering the extra objective function Kj , m − h + 1 candidate values b¯ (j0 ) , b¯ (j1 ) , . . . , b¯ (jm−h ) are obtained for b∗j . Let h+ p h   ∗ K j( p ) = (rˆ(i, j ) − 1 + aˆi )2 + (rˆ(i, j ) − aˆi )2 + i=1

i=h+1

m 

(rˆ(i, j ) − b¯ (j p) )2

i=h+ p+1

In this way, ∗ ∗ ∗ min K j( p ) , q∗ = arg min K j( p ) and b∗j = bqj .

K ∗j =

0≤p≤m−h

∗ = Therefore, Kop



0≤p≤m−h

n j=1

K ∗j and the fuzzy consequence based on Kop is B∗ = [b∗1 , b∗2 , . . . , b∗n ].

(c) Considering the extra objective function Lj , m − h + 1 candidate values b˜ (j0 ) , b˜ (j1 ) , . . . , b˜ (jm−h ) are obtained for b∗j . Let ∗ L(j p ) = max



max |rˆ(i, j ) − 1 + aˆi |,

1≤i≤h

max

h+1≤i≤h+ p

|rˆ(i, j ) − aˆi |,

max

h+ p+1≤i≤m

 |rˆ(i, j ) − b˜ (j p) | .

In this way,

L∗j =

∗ ∗ ∗ min L(j p ) , q∗ = arg min L(j p ) and b∗j = bqj .

0≤p≤m−h

0≤p≤m−h

Therefore, L∗op = max L∗j and the fuzzy consequence based on Lop is B∗ = [b∗1 , b∗2 , . . . , b∗n ]. 1≤ j≤n

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

667

4.4. The consistency of the proposed OFR methods The reasoning consistency is a basic characteristic for fuzzy reasoning methods. It means that for the fuzzy rule “If X is A, then Y is B” and the given fuzzy premise A, the resulting fuzzy consequence B∗ is equal to B. It was proved that the OFR method based on IM satisfies reasoning consistency [24]. Now we discuss the reasoning consistency for the OFR methods based on IL , IK and IZ . Proposition 3. Suppose the fuzzy relation R is gained from the fuzzy rule “If X is A, then Y is B”. Let the fuzzy premise be given as A = [ai ]m×1 such that A is normal (i.e., max ai = 1), then the OFR methods based on IL , IK and IZ satisfy reasoning consistency. 1≤i≤m

Proof. Let B = [b j ]1×n and

⎡

B∗

=

[b∗j ]1×n .

I ( a1 , b1 ) ⎢ I ( a2 , b1 ) R=A→B=⎢ .. ⎣ . I ( am , b1 ) we have

⎡

I (a1 , b∗1 ) I (a2 , b∗1 ) ⎢ R = A → B∗ = ⎢ .. ⎣ . I (am , b∗1 )

Since

I ( a1 , b2 ) I ( a2 , b2 ) .. . I ( am , b2 ) I (a1 , b∗2 ) I (a2 , b∗2 ) .. . I (am , b∗2 )

... ... .. . ... ... ... .. . ...

⎤

I ( a1 , bn ) I ( a2 , bn ) ⎥ ⎥, .. ⎦ . I ( am , bn )

⎤

I (a1 , b∗n ) I (a2 , b∗n ) ⎥ ⎥. .. ⎦ . I (am , b∗n )

Considering that [ai ]m × 1 is normal, without loss of generality, we let a1 = 1. Then the distance between R and R is

⎡

I (a1 , b∗1 ) − I (a1 , b1 ) ⎢ I (a2 , b∗1 ) − I (a2 , b1 ) R − R = ⎢ .. ⎣ . I (am , b∗1 ) − I (am , b1 )

I (a1 , b∗2 ) − I (a1 , b2 ) I (a2 , b∗2 ) − I (a2 , b2 ) .. . I (am , b∗2 ) − I (am , b2 )

⎤

... ... .. . ...

I (a1 , b∗n ) − I (a1 , bn ) I (a2 , b∗n ) − I (a2 , bn ) ⎥ ⎥. .. ⎦ . ∗ I ( am , bn ) − I ( am , bn )

... ... .. . ...

b∗n − bn I (a2 , b∗n ) − I (a2 , bn ) ⎥ ⎥, .. ⎦ . I (am , b∗n ) − I (am , bn )

Since IL , IK and IZ satisfy I (1, y ) = y for all y ∈ [0, 1], we have

⎡

b∗1 − b1 I (a2 , b∗1 ) − I (a2 , b1 ) ⎢ R − R = ⎢ .. ⎣ . I (am , b∗1 ) − I (am , b1 )

b∗2 − b2 I (a2 , b∗2 ) − I (a2 , b2 ) .. . I (am , b∗2 ) − I (am , b2 )

⎤

where I = IL , IK and IZ . Let us use the objective functions Jop , Kop and Lop to measure R − R. Then we have Jop , Kop , Lop = 0 in case b∗j = b j ( j = 1, 2, . . . , n ). Suppose there exists j0 such that b∗j = b j0 , we have Jop , Kop , Lop > 0. That is to say, Jop , Kop and 0

Lop achieve their minimum at [bj ]1 × n . Thus the fuzzy consequence B∗ = [b j ]1×n = B.



5. The relationship between OFR methods and POFI methods In [13,14], Peng et al. proposed a fuzzy reasoning method called the pointwise optimizing fuzzy inference (POFI) method. However, they did not present the way of calculating the POFI solutions. In this section, we prove that the POFI method is a special case of the OFR method. Then we obtain the POFI solutions through the computational process of OFR methods. Note that the FMP can be represented as Suppose that and given

A1 → B1 A

calculate

B∗

where A1 and A are fuzzy sets defined on the universe of discourse U and B1 and B∗ are fuzzy sets defined on the universe of discourse V. Let us regard the proposition A1 → B1 as a fuzzy relation I (μA1 (u ), μB1 (v )), where I is a fuzzy implication. The idea of POFI method for FMP for a given u1 ∈ U is to determine the fuzzy set B∗ defined on the universe of discourse V such that the formula

max |I (μA1 (u1 ), μB1 (v j )) − I (μA (u1 ), μB∗ (v j ))| v j ∈V

(5)

reaches its minimum value at u1 ∈ U. Suppose U and V are finite discrete universes with U = {u1 , u2 , . . . , um } and V = {v1 , v2 , . . . , vn }. In the OFR method, we use [r(i, j)]m × n to represent the given fuzzy relation. The given fuzzy premise and the fuzzy consequence are denoted as [a1 , a2 , . . . , am ] and [b∗1 , b∗2 , . . . , b∗n ] with ai = μA (ui ) and μB∗ (v j ). Therefore, the symbols I (μA1 (u1 ), μB1 (v j )), μA (u1 ) and

668

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

μB∗ (v j ) of POFI method equate to the symbols r(1, j), a1 and b∗j of OFR method. The reasoning goal of POFI method is to determine [b∗1 , b∗2 , . . . , b∗n ] such that the objective function

Mop = max |r (1, j ) − I (a1 , b∗j )| 1≤ j≤n

is minimized. It is shown that the objective function Lop in the OFR method is Lop =

max

1≤i≤m,1≤ j≤n

|r (i, j ) − I (ai , b∗j )|. If m = 1,

then Lop = max |r (1, j ) − I (a1 , b∗j )| = Mop . That is to say, if m = 1, then the OFR method is equal to the POFI method. In 1≤ j≤n

[13], Peng et al. proposed POFI solutions of (5) for IM and IZ . However, they did not present the way of calculating the POFI solutions. Considering that the POFI method is a special case of the OFR method, determining B∗ that minimizes formula (5) in the POFI method can be converted to calculating the fuzzy consequence [b∗1 , b∗2 , . . . , b∗n ] of the OFR method. Proposition 4. Let the fuzzy premise A = [ai ]1×m and the fuzzy relation R = [r (i, j )]m×n . Considering the OFR methods based on objective functions Jop , Kop and Lop and different fuzzy implications I, we obtain the following results for m = 1. (1) (2) (3) (4)

If If If If

I = IM , then b∗j = min(r (1, j ), a1 ). I = IL , then b∗j = max(r (1, j ) − 1 + a1 , 0 ). I = IK , then b∗j = max(r (1, j ), 1 − a1 ). I = IZ , then b∗j = min(r (1, j ), a1 ).

Proof. It is straightforward from the computational process of OFR methods based on IM , IL , IK and IZ .



Remark 4. (1) For the OFR method based on IZ , if 21 ≤ a1 ≤ 1, then b∗j = min(r (1, j ), a1 ). If a1 ≤ 12 , then b∗j can take arbitrary value in the interval [0, 1]. Specially, let b∗j = min(r (1, j ), a1 ). Thus the fuzzy consequence for the OFR method based on IZ can be uniformly expressed as min (r(1, j), a1 ). (2) If m = 1, then the OFR methods based on objective functions Jop , Kop and Lop are equivalent no matter I is IM or IL or IK or IZ . The following results for the POFI method can directly be obtained from Proposition 4. Proposition 5. If I = IM , then the min(min(μA1 (u1 ), μB1 (v j )), μA (u1 )).

POFI

solution

for

formula

(5)

can

be

expressed

as

μB ∗ ( v j ) =

Proposition 6. If I = IL , then the POFI solution for formula (5) can be expressed as μB∗ (v j ) = max(min(μB1 (v j ) − μA1 (u1 ) + μA (u1 ), μA (u1 )), 0 ). POFI

solution

for

formula

(5)

can

be

expressed

as

μB∗ (v j ) = max(max(1 −

Proposition 8. If I = IZ , then the POFI μA1 (u1 ), min(μA1 (u1 ), μB1 (v j ))), μA (u1 )).

solution

for

formula

(5)

can

be

expressed

as

μB∗ (v j ) = min(max(1 −

Proposition 7. If I = IK , then μA1 (u1 ), μB1 (v j )), 1 − μA (u1 )).

the

Remark 5. The results obtained in Propositions 5–8 are in accord with those obtained in [13,14]. 6. Conclusions The OFR method is a new fuzzy reasoning method that can be used for modeling and control of complex systems and for decision-making under complex environments [4,9]. Based on the existing frameworks of OFR methods [4,23], we made some notes on OFR methods in this paper. The main contribution of this paper is described in details as follows. First, the OFR method based on the objective function Lop , which is proposed in [23], was modified since the original computational process of this method was not complete. Second, it was pointed out in the conclusions of [23] that the analysis of the advantages and limitations of OFR methods deserved investigation. It is still an open problem now. Thus some advantages and limitations of OFR methods were examined in the present paper. Third, considering that the Łukasiewicz implication, the Kleene-Dienes implication and the Zadeh implication are the most robust operators among R-implications, S-implications and QL-implications, respectively, we applied them to OFR methods and obtained new OFR methods. It was proved that the proposed new OFR methods satisfy reasoning consistency. Finally, the relationship between the OFR method and the POFI method was examined. It was shown that POFI methods are special cases of OFR methods. Thus determining POFI solutions of POFI methods can be converted to calculating fuzzy consequences of OFR methods. A summary of the referred work and our proposed work about OFR methods can be found in Table 2. The following is some corresponding work that deserves investigation in our further research. First, the ways to overcome the limitations of OFR methods should be sought. Second, the robustness of the proposed new OFR methods should be examined. Third, the comparison on the effect of different OFR methods should be made through practise examples and the impact of different fuzzy implications on OFR methods should be analyzed.

Y. Li, X. He and K. Qin et al. / Information Sciences 503 (2019) 652–669

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Table 2 A summary of the referred work and our proposed work about OFR methods. The main contribution

Zhang and Cai’s work

The present work

The implications The objective functions The OFR methods The comparison between the OFR method and the CRI method

IM Jop , Kop and Lop based on IM The OFR method based on IM It was shown in [4] that the OFR method can behave better in terms of robustness than the CRI method. It was proved in [23] that the OFR method based on IM satisfies consistency.

IL , IK , IZ Jop , Kop and Lop based on IL , IK and IZ The OFR methods based on IL , IK and IZ (Sections 4.1–4.3) The advantages and limitations of OFR methods are illustrated. It is proved that OFR and CRI methods are equivalent under certain conditions. (Section 3.2) It is proved that the OFR methods based on IL , IK and IZ satisfy consistency. (Section 4.4)

The consistency of OFR methods

Other special work in the present work

It is proved that Zhang and Cai’s OFR method based on Lop is not correct and thus we give a modification of it. (Section 3.1) It is proved that POFI methods proposed in [13,14] are special cases of OFR methods. (Section 5)

Conflict of interest The authors declare that they have no conflict of interest to this work. Acknowledgments The authors are very grateful to the Editor and referees, for their constructive comments and useful suggestions that help us to improve the quality of the paper. This research is supported by the National Natural Science Foundation of China (Grant Nos. 61603307, 61473239, 71301129 and 61305074), the Fundamental Research Funds for the Central Universities (Grant Nos. JBK1801078 and JBK150503) and the Grant from MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Grant No. 19YJCZH048). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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