Quantifiers induced by subjective expected value of sample information with Bernstein polynomials

European Journal of Operational Research 254 (2016) 226–235

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Quantifiers induced by subjective expected value of sample information with Bernstein polynomials Kaihong Guo∗ School of Information, Liaoning University, Shenyang 110036, China

a r t i c l e

i n f o

Article history: Received 25 August 2014 Accepted 11 March 2016 Available online 19 March 2016 Keywords: Uncertainty modeling Personalized quantifier Bernstein polynomials Ordered weighted averaging (OWA) aggregation

a b s t r a c t A kind of personalized quantifier, the so-called SEVSI-induced quantifier as an acronym for Subjective Expected Value of Sample Information, is developed in this paper by introducing Bernstein polynomials of higher degree. This allows us to provide a novel solution to improve the final representation of the quantifier that generally performed poorly in our previous work, thus enhancing the quality of global approximation of functions and improving the operability of this kind of quantifier for practical use. We show some properties of the developed quantifier. We also prove the consistency of the OWA aggregation under the guidance of this type of quantifier. Finally, we experimentally show that the developed quantifier outperforms the one with the piecewise linear interpolation in many aspects of geometrical characteristics and operability. Thus it could be considered as an effective analytical tool to help handle the complex cases involving people’s personalities or behavior intentions that have to be considered in decision making under uncertainty. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In natural language there are many linguistic quantifiers exemplified by terms such as more than 10, most, some, few, and about half. So far there have been several attempts to deal with this topic, notable among these are the work by Liu (2005), Liu and Han (20 08), Yager (20 04, 1996), and Zadeh (1983, 1965). And a detailed overview can be found in Guo (2014). This paper is a direct continuation of our previous work (Guo, 2014) in which we proposed a kind of personalized quantifier, the so-called SEVSI-induced one as an acronym for Subjective Expected Value of Sample Information, which can be associated directly with a specified decision maker (DM) and used as a tool to investigate and formalize his/her decision attitude or behavior intention. This makes us believe that it has a wide range of applications in increasingly complex situations. In particular, it may help to bring about more intuitively appealing and convincing results in decision making under uncertainty, especially in the ordered weighted averaging (OWA) aggregation under the guidance of quantifier. Noted that the quantifier was realized in Guo (2014) by the piecewise linear interpolation and represented as a piecewise linear function, of which the number of pieces depends on the number of the alternatives in the given sample. Theoretically, the more pieces of the piecewise

∗

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http://dx.doi.org/10.1016/j.ejor.2016.03.015 0377-2217/© 2016 Elsevier B.V. All rights reserved.

function, the better performance of the global approximation by this interpolation. For practical applications, however, too many pieces may throw users into confusion and such piecewise functions may become difficult to handle. Besides, the smoothness of the fitted curves by this interpolation may still need improving. Hence, more efforts should be made to tackle these issues so as to further perfect this kind of quantifier. In this paper, we develop a new type of SEVSI-induced quantifier by introducing Bernstein polynomials of higher degree, thus providing a novel solution to the unsolved problems mentioned above. The consistency of the OWA aggregation under the guidance of the developed quantifier is also addressed and proved. Our aim is to develop a kind of personalized quantifier with excellent properties and pleasing operability, so as to provide an effective analytical tool with a sound theoretical basis for practical applications in more complex situations. Since the Regular Increasing Monotone (RIM) quantifier is a basis for constructing another two kinds of relative quantifiers, namely the Regular Decreasing Monotone (RDM) quantifier and the Regular UniModal (RUM) quantifier (Yager, 1996), all of the quantifiers considered in this paper are assumed to be RIM. The rest of this paper is organized as follows: Section 2 briefly recalls the OWA operator and Bernstein polynomials. In Section 3, the SEVSI-induced quantifier with Bernstein polynomials of higher degree is investigated in considerable detail. Section 4 makes use of two numerical examples to illustrate and examine the developed quantifier, followed by conclusions in Section 5.

K. Guo / European Journal of Operational Research 254 (2016) 226–235

2. Preliminaries

case, let

2.1. OWA operators

λQ = lim AC (Q ) = n→∞

Yager (1988) introduced the concept of the OWA operator that is defined as follows. The OWA operator of dimension n is a mapping F : Rn → R with an associated weighting vector W = (w1 , w2 , . . . , wn )T such that

FW (x1 , x2 , . . . , xn ) =

n 

w jy j,

(1)

FW (x1 , x2 , . . . , xn ) = W T Y, where Y = (y1 , y2 , . . . , yn )T is called an OWA argument vector. It is well known that an OWA weighting vector plays a key role in the aggregation process (Ahn, 2011, Bustincea, Fernandeza, Kolesárováb, & Mesiar, 2015). Yager (2009), Yager (2004), Yager (1996) suggested an effective approach to obtain an OWA weighting vector via the RIM quantifiers that can be denoted by a fuzzy subset Q with the following properties: 1) Q (0 ) = 0, 2) Q (1 ) = 1, 3)Q (x ) ≥ Q (y )if x ≥ y. These quantifiers were denoted as basic unit-interval monotonic (BUM) functions in Yager (2004). Using a quantifier Q, we can obtain the OWA weights as

  j n

wj = Q





j−1 , n

−Q

j = 1, 2, . . . , n n

It is clear that w j ∈ [0, 1] and rewritten as

j=1

w j = 1. Eq. (1) can then be

FQ (x1 , x2 , . . . , xn ) = FW (x1 , x2 , . . . , xn ) =

n 

   Q

j=1

j n



−Q

(2)

j−1 n

AC (W ) =

j=1

n− j wj . n−1

 y j.

(3)

(4)

It can be shown that AC (W ) ∈ [0, 1]. Specifically, larger values of AC (W ), closer to 1, are an indication of preference for larger argument values in the aggregation. While lower values of AC (W ), closer to 0, are an indication of preference for smaller argument values in the aggregation. Values of AC (W ) in the middle, near 0.5, can be a reflection of no preference for either large or small argument values at all. Given a connection between Eq. (2) and Eq. (4), the measure of attitudinal character can be rewritten as

AC (Q ) =

n 

   Q

j=1

j n



−Q

j−1 n



n− j . n−1

(5)

Further algebraic manipulation of the formula leads to the following simple form

AC (Q ) =

1 n−1

n−1  j=1

 

Q

j . n

Q (x )dx.

(7)

Recall the concept of classical Bernstein polynomials that is defined as follows. Let f (x ) ∈ C[0, 1]. A sequence of Bernstein polynomials of f is (Cheney, 1982)

Bn ( f ; x ) =

n 

 

f

k=0

k n−k C k xk ( 1 − x ) , n n

(8) n! , and the k!(n−k )! k k Cn x (1 − x )n−k (k =

where the binomial coefficients are given by Cnk = Bkn (x )

Bernstein basic functions are defined by = 0, 1, . . . , n ). According to Bernstein’s proof of the Weierstrass Approximation Theorem, lim Bn ( f ; x ) = f (x ) for any function f (x ) ∈ n→∞

C[0, 1] (Cheney, 1982). Bernstein polynomials have some best properties among all approximating polynomials (Ghosh & Osman, 2012), some of which are briefly introduced as follows. 1) Normalization of the Bernstein basic functions, i.e., n k k k=0 Bn (x ) = 1 where Bn (x ) ≥ 0 for any x ∈ [0, 1]. Proof. Trivial from the binomial expansion.



2) End-point interpolation, i.e., Bn ( f ; 0 ) = f (0 ), and Bn ( f ; 1 ) = f ( 1 ). Proof. Trivial from the algebraic manipulation of Eq. (8).



3) Positive linear operator, i.e., for any f (x ), g(x ) ∈ C[0, 1], Bn ( f + g; x ) = Bn ( f ; x ) + Bn (g; x ), Bn (a f ; x ) = aBn ( f ; x ) where a ∈ R, and Bn ( f ; x ) ≥ 0 if f (x ) ≥ 0.

This is called the quantifier guided OWA aggregation (Yager, 1996). Yager (1993), Yager (1988) further introduced a characterizing measure called the attitudinal character that can be used to characterize an OWA weighting vector with respect to any distinction between preferences for large or small argument values. The attitudinal character is defined as n 

0

1

2.2. Bernstein polynomials

j=1

n where w j ∈ [0, 1], j=1 w j = 1, and y j is the j th largest of xi (i = 1, 2, . . . , n ). With the help of vector notations, Eq. (1) can be expressed as



227

(6)

Obviously, if n → +∞, then the concept of attitudinal character can be associated directly with a quantifier Q (Yager, 2004). In this

Proof. Trivial from the algebraic manipulation of Eq. (8).



Thus it can be easily deduced that Bn ( f ; x ) ≥ Bn (g; x ) if f (x ) ≥ g(x ) for any x ∈ [0, 1]. 4) Linear invariant, i.e., for any linear function l (x ) = ax + b where a, b ∈ R, Bn (l; x ) = l (x ). Proof. It is clear from the property 3) that Bn (ax + b; x ) = aBn (x; x ) + bBn (1; x ), where

Bn ( 1; x ) =

n 

n−k

Cnk xk (1 − x )

= (x + (1 − x ))n = 1,

k=0

and

Bn ( x; x ) =

n n   k k k k k k n−k n−k Cn x (1 − x ) = C x (1 − x ) n n n k=0

k=1

n−1  k + 1 k+1 k+1 n−k−1 = Cn x (1 − x ) n k=0

=x

n−1 

n−k−1

Cnk−1 xk (1 − x )

= x(x + 1 − x )n−1 = x.

k=0

Thus Bn (ax + b; x ) = aBn (x; x ) + bBn (1; x ) = ax + b, i.e., Bn (l; x ) = l ( x ).  This means all of the linear functions are the fixed points of Bn . 5) Geometrically shape-preserving property, i.e., Bn ( f ; x ) is monotonically increasing (or convex) over [0, 1] if the function f (x ) ∈ C[0, 1] does so. Proof. (a) Assume that f (x ) ∈ C[0, 1] is monotonically increasing (or decreasing) over [0, 1]. We shall prove Bn ( f ; x ) does so by

228

K. Guo / European Journal of Operational Research 254 (2016) 226–235

checking the sign of its first derivative over [0, 1]. Note that n−k 

( x (1 − x ) k

Thus,

B n ( f ; x ) =

) = kx

f

n 

f

f

n−1 

n−1 

1 f 2

f

 

1 f 2

1 0

f

f

  −f

k n

Cnk−1 xk

n−k−1

(1 − x )

.

f

k+1 n



k n

n

( f ; x) = n

n−1 

  f

k+1 n



= =

−f n−k−1

× [kxk−1 (1 − x ) n−1 

  f

k=1

k+1 n

  n−2

−n

 k=0

f



k+1 n

Cnk−1

=n

n−k−2

  −f

k n



  −f

k=0

×

+1 Cnk−1

f

k+2 n

k n

k+1 n

−f

n−k−2

( k + 1 )x ( 1 − x ) k

  +f

k n

k n

 

≥ f



1 k+2 k + 2 n n k+1 n





⇒ Bn ( f ; x ) ≥ 0,

  k n

+f

 

≤ f



1 k+2 k + 2 n n k+1 n





⇒ Bn ( f ; x ) ≤ 0,

0

]



n  

1 0

1 0

Bkn (x )dx(k = 0, 1, . . . , n). It is clear that Ink ≥ 0(k =

Bkn (x )dx =

1

Bkn (x )dx = Cnk

1 k+1



1



1

0

 0

1

n 

Bkn (x )dx = 1.

k=0

n−k

xk ( 1 − x )

dx

(1 − x )n−k dxk+1

   1 1 n−k k+1 1 n−k k+1 [ ( 1 − x ) x ]0 − x d (1 − x ) k+1 0  1 1 Cnk (n − k )xk+1 (1 − x )n−k−1 dx k+1 0  n − k 1 k+1 n−k−1 Cnk x (1 − x ) dx k+1 0  1  1 n−k−1 Cnk+1 xk+1 (1 − x ) dx = Bkn+1 (x )dx = Ink+1 . 0

Cnk

0

0

Ink

Ink+1

That means = for all k = 0, 1, . . . , n − 1. Thus Ink = 1 k Bn (x )dx = n+1 holds for all k = 0, 1, . . . , n. 

3. SEVSI-induced quantifier with Bernstein polynomials

n−k−1

Cnk−1 kxk−1 (1 − x )

Cnk−1



= 1

− ( n − k − 1 )xk ( 1 − x )

(n − k − 1 )xk (1 − x )n−k−2      n−2

×

=

  k n

0

= Cnk

≤ 0 ⇒ Bn ( f ; x ) ≤ 0,

k=0

=n



k=0



≥ 0 ⇒ Bn ( f ; x ) ≥ 0,

k n

Ink =

Ink =

which implies Bn ( f ; x ) is monotonically decreasing over [0, 1], too. (b) Assume f (x ) ∈ C[0, 1] is convex (or concave) over [0, 1]. We shall prove Bn ( f ; x ) does so by checking the sign of its second derivative over [0, 1], that is,

B

 

+f

Note that

 

−f

k+2 n

k=0

which implies Bn ( f ; x ) is also monotonically increasing over [0, 1]. On the other hand, when f (x ) is monotonically decreasing over [0, 1], then







6) Integral property, i.e., for any Bernstein basic function Bkn (x ), 1 Bkn (x )dx = n+1 for all k = 0, 1, . . . , n.

n 

 

−f

k+2 n

Proof. Let Ink = 0, 1, . . . , n) and

k n−k−1 C k xk ( 1 − x ) n n−1



− 2f

k+1 n

which implies Bn ( f ; x ) is concave over [0, 1], too.

 

k+1 n



(1 − x )n−k−2

= f

k+1 n−k−1 Cnk−1 xk (1 − x ) n

 

f



which implies Bn ( f ; x ) is also convex over [0, 1]. On the other hand, when f (x ) is concave over [0, 1], then



f

k n

−f

k+2 n

= f

k n−k−1 C k ( n − k )xk ( 1 − x ) n n

n−1 

n−1 





  

 

When f (x ) is convex over [0, 1], then

 

When f (x ) is monotonically increasing over [0, 1], then

k+1 n

Cnk−2 xk

×

k+1 n−k−1 Cnk+1 (k + 1 )xk (1 − x ) n

k=0



= n (n − 1 )

  n−2 k=0

k=0

=n





k=0

−n

n−k

k n−k−1 C k ( n − k )xk ( 1 − x ) n n

f



n−k−2

  f



k=0

=n

k+1 n

k n−k C k kxk−1 (1 − x ) n n

k=0

−

f

k=0

 

f

k=0

=

 

× Cnk−1 (n − k − 1 )xk (1 − x )

k n−k−1 C k ( n − k )xk ( 1 − x ) n n

f

n−1 

n−1 

.

 

k=1

−

−n

k n−k C k kxk−1 (1 − x ) n n

k=0 n 

n−k−1

 

n 

−

− ( n − k )x ( 1 − x ) k

Cn xk (1 − x )

n

k=0

=

(1 − x )

n−k

  n  k k k=0

=

k−1

n−2 

This section is partially based on an interactive testing process that was developed in our previous work (Guo, 2014) to extract from a specified DM information about his/her decision attitude or behavior intention with sample information. Without loss of generality, the set of sample information is assumed to be a multi-attribute decision matrix with m alternatives and n criteria, denoted by D = (di j )m×n where di j ∈ R. Let D¯ = (d¯i j )m×n be the normalization of D = (di j )m×n where d¯i j ∈ [0, 1]. Note that the values of each criterion in D¯ = (d¯i j )m×n are rearranged in descending order in terms of the idea of OWA aggregation. The specified DM

K. Guo / European Journal of Operational Research 254 (2016) 226–235

is then asked, based on personal preference or decision attitude, to provide his/her expected or satisfactory value for each criterion with respect to D¯ . The set of subjective expected values of criteria provided by this DM can be expressed via a vector, denoted by ν¯ = (v¯ 1 , v¯ 2 , . . . , v¯ n )T where v¯ j ∈ [0, 1]( j = 1, 2, . . . , n). Thus the information can be extracted, with the help of D¯ and ν¯ , from this DM about his/her decision attitude or behavior intention, which is denoted by an attitudinal weighting vector (an OWA weighting vector in nature), S = (s1 , s2 , . . . , sm )T , where (Guo, 2014)

1

s0 = 0, si =  n

2 m ¯ ¯ j) j=1 (di j − v k=1

n j=1

,

1 2 (d¯k j −v¯ j )

i = 1, 2, . . . , m. (9)

 It can be shown that si ≥ 0(i = 0, 1, . . . , m) and m i=0 si = 1.  According to Eq. (2), let Q ( mi ) = ik=0 sk for all i = 0, 1, . . . , m where Q represents an undetermined quantifier associated directly with this DM. In this way only some discrete values of Q are determined where Q (0 ) = 0, Q (1 ) = 1. Based on these values, more efforts are made in the following to fit a continuous and smooth function over [0, 1] with the Lagrange interpolation and Bernstein polynomials, respectively. 3.1. Try of Lagrange interpolation

m 

li ( x )Q ( xi ) =

i=0

m m  x − xj Q (xi ), x ∈ [0, 1], xi − x j

(10)

i=0 j=0 j =i

where the interpolating basic functions are given by

li ( x ) =

m m m x − xj x − jh mx − j = = , xi − x j ih − jh i− j j=0 j =i

j=0 j =i

x ∈ [0, 1],

Given the polynomial integral in Eq. (13), the calculations of Ci(m ) (i = 0, 1, . . . , m) do not get into real trouble. However, the nonnegativity of Ci(m ) cannot be ensured for all i = 0, 1, . . . , m with the increasing of m. Indeed, there is Ci(m ) < 0 when m ≥ 8 (Cheney, 1982). This may lead to the computational instability of the attitudinal character λQL,m . Moreover, with the degree m getting higher, the fitting function QL,m (x ) may suffer from serious oscillation and larger errors of computation, known as the Runge’s phenomenon. Therefore, the use of the Lagrange interpolation polynomial of higher degree, especially of degree more than 8, is not meant to be considered for practical applications. 3.2. Use of Bernstein polynomials Let’s now try another type of polynomial approximation with Bernstein polynomials. From the structure of Eq. (8), our personalized quantifier can be defined by

QB,m (x ) = Bm (Q; x ) =

j=0 j =i

QL,m (x ) =

i=0 j=0 j =i

1 0

QB,m (x )dx =

 

i , m

x ∈ [0, 1].

=

(11)

λQL,m =

1 0

QL,m (x )dx =

1 m

i m

Q

i=0

Let Ci(m ) =

Ci(m ) =

 



m 0

m 1 0

j=0 j =i

mx− j dx(i i− j

m 1

0

j=0 j =i

mx − j dx. i− j

j=0 j =i

= (12)

λQL,m =

i=0

 

Q

i m



m 0

m

Q

i m−i C i xi (1 − x ) dx m m

 

Q

i m

1 0

m 1  Q m+1

1 m+1

Bim dx

 

m  i 

i m

sk =

i=1 k=1



m 

m 1  ( m − i + 1 )si m+1 i=1

( m + 1 )si −

i=1

m 



isi

i=1

m 1  isi . m+1 i=1

(13)

Thus, the attitudinal character associated with QB,m can be measured by

λQB,m = (t − j )dt .

 

i=0

= 1−

j=0 j =i

(−1 )m−i m ( m − i )! i !

0 m 

1 = m+1

thus Eq. (12) can be rewritten as m 

m 1

i=1

= 0, 1, . . . , m) and x = th. Then,

 m m m t−j (−1 )m−i dt = (t − j )dt , i− j m ( m − i )! i ! 0



i=1

=

m 

(15)

Q (x ) ∈ C[0, 1]. Thus, the fitting function QB,m (x ) is characterized by many excellent properties of Bernstein polynomials that may ensure the high performance of global approximation. Obviously, the function QB,m (x ) is monotonically increasing over [0, 1] as the undetermined quantifier Q (x ) is assumed to be RIM. In other words, QB,m (x ) ≥ QB,m (y )if x ≥ y for any x, y ∈ [0, 1]. In addition, according to the aforementioned end-point interpolation property, it is easy to understand that QB,m (0 ) = Bm (Q; 0 ) = Q (0 ) = 0 and QB,m (1 ) = Bm (Q ; 1 ) = Q (1 ) = 1. Therefore, our personalized quantifier QB,m is also denoted by a BUM function. Let’s first investigate the attitudinal character associated with QB,m . Note that

Thus the attitudinal character associated directly with the quantifier QL,m is defined by



i m−i C i xi (1 − x ) , x ∈ [0, 1], m m

Q

m! i = where the binomial coefficients are given by Cm . Aci!(m−i )! cording to Bernstein’s theorem on approximation of function, it is certain that lim QB,m (x ) = Q (x ) for any undetermined quantifier

Eq. (10) can then be rewritten as

mx − j Q i− j

 

i=0



i = 0, 1, . . . , m. m m 

m 

m→∞

1 Let the step length h = m and the isometric nodes xi = ih(i = 0, 1, . . . , m). In this way, our personalized quantifier can be fitted as a Lagrange interpolation polynomial of degree m, i.e.,

QL,m (x ) =

229

(14)

j=0 j =i

It is worth noticing that the notation Ci(m ) in Eq. (13) is m (m ) called the Cotes coefficient where = 1 (Cheney, 1982). i=0 Ci



1 0

QB,m (x )dx = 1 −

m 1  isi , m+1

(16)

i=1

where si (i = 1, 2, . . . , m) are given by Eq. (9). Some special cases are worth pointing out. As the involved DM gives more preference for the larger sample values with higher SEVSI, then the attitudinal weighting vector S → (1, 0, . . . , 0 )T . In this case, it is clear that λQB,m = 1 − 1/(m + 1 ). When m → +∞, then λQB,m = 1.

230

K. Guo / European Journal of Operational Research 254 (2016) 226–235

This is clearly an optimistic attitude. On the other hand, by giving more preference for the smaller sample values with lower SEVSI, the DM may provide such a vector S → (0, 0, . . . , 1 )T with λQB,m = 1 − m/(m + 1 ). When m → +∞, then λQB,m = 0. This is clearly pessimistic. If the DM gives a preference for the sample values in the middle between the large and small ones, then the significant weights will be dispersed as evenly as possible around the middle positions in S. In this case, λQB,m ≈ 0.5, which is obviously an indi-

(1/m, 1/m, . . . , 1/m )T ,

cation of neutral attitude. When S =

λQB,m

then

FQB,m (X ) =

=

  j n

QB,m =

m 

1 = 1− (sm + 2sm−1 + 3sm−2 + · · · + ms1 ) m+1 m m 1  1  = 1− ism−i+1 = 1 − ( m − i + 1 )si m+1 m+1 1 = 1− (m + 1 ) m+1



= 1−

si −

i=1



m

=

1 m+1

1

n−1 

j n

xj

j=1



PB,m ( nj )(x j − x j+1 ) + xn .

  j n

QB,m

We

then

  − PB,m

j n

(x j − x j+1 ). (17)

i −P m

j n

i Cm

j 1− n

m−i ≥ 0.

0 m 

=

  Q

i=1 m 

=

i m

1

0

  Q

i=1

i m

1

0

m 1  = Q m+1 i=1

Similarly, λPB,m =



m 1

0

 

i m−i C i xi (1 − x ) dx m m

Q

i=1

m−i

i i Cm x (1 − x )

dx

Bim (x )dx

 

i . m

1

  i m

PB,m (x )dx =

m

i i i=1 P ( m ). Since Q ( m ) ≥   m i i P ( mi ) for all i = 1, 2, . . . , m, which implies m i=1 Q ( m ) ≥ i=1 P ( m ), we further get 0

1 m+1

 

m i 1  ≥ P , m+1 m i=1

i.e., λQB,m ≥ λPB,m .

isi



i Cm

i xi (1 − x )m−i (i = 1, 2, . . . , m) 3) Let Bim (x ) = Cm where m! = i!(m−i )! and x ∈ [0, 1]. It is certain that Bim (x ) ≥ 0 for all

i = 1, 2, . . . , m. Since Q ( mi ) ≥ P ( mi ) for all i = 1, 2, . . . , m, then

 

Q

i=1

 

i i Bim (x ) ≥ P Bim (x ), i = 1, 2, . . . , m, x ∈ [0, 1], m m

which implies 

m 

Theorem 2. Let QB,m and PB,m be two SEVSI-induced quantifiers with Bernstein polynomials generated respectively from q q q two attitudinal weighting vectors Sq = (s1 , s2 , . . . , sm )T and S p = p T (s1p , s2p , . . . , sm ) where sqi , sip (i = 1, 2, . . . , m) are given by Eq. (9). Let λQB,m and λPB,m be two attitudinal characters associated with  q QB,m and PB,m , respectively. For Sq and S p , let Q ( mi ) = ik=1 sk  p and P ( mi ) = ik=1 sk (i = 1, 2, . . . , m). If Q ( mi ) ≥ P ( mi ) for all i = 1, 2, . . . , m, then 1) FQB,m (X ) ≥ FPB,m (X ) for any argument vector

i=1

X = (x1 , x2 , . . . , xn )T ; 2) λQB,m ≥ λPB,m ; and 3) QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1].

)T ,

n−1

   i 

QB,m (x )dx =

i=1

isi = 1 − λQB,m .

Proof. 1) For X = (x1 , x2 , . . . , xn xn . From Eq. (3), we have



λQB,m =

i=1

m 



Given x j ≥ x j+1 for all j = 1, 2, . . . , n − 1, it is certain that FQB,m (X ) ≥ FPB,m (X ). 2) Note that

i=1

 1 (m + 1 ) − isi m+1

i m

m 1  Q m+1

i=1

m 

   Q

− QB,m

j−1 n

 

− PB,m

i=1

B,m

Proof. From Eq. (16) and S¯ = (sm , sm−1 , . . . , s1 )T , it is easy to understand that

m 

QB,m



j (x j − x j+1 ) + xn . n

QB,m

Note that

Theorem 1. Let QB,m be a SEVSI-induced quantifier with Bernstein polynomials generated from an attitudinal weighting vector S = (s1 , s2 , . . . , sm )T where si (i = 1, 2, . . . , m) are given by Eq. (9). Let λQB,m ∈ [0, 1] be the attitudinal character associated with QB,m . If the quantifier Q¯ B,m is generated from the reverse order of S, S¯ = (sm , sm−1 , . . . , s1 )T , then λ ¯ = 1 − λQ .

i=1

j n

j=1

which is an indication of complete neutrality. As some extreme cases, let us see what happens if the DM intends to give his/her preference at random, or the sample data are not representative enough to be used to extract correct attitudinal information. Given the manner in which the SEVSI is provided, it is clear that the vector S and the resulting quantifier QB,m may show a presentation of some kind of overall evaluation on the performance by this DM. Accordingly, the value of λQB,m in this case also gives a similar indication that is actually difficult to explain with logical arguments. We now look at the consistency of the OWA aggregation guided by QB,m .



 

 

FPB,m (X ) =

Similarly, calculate



j=1

j=1

i=1

λQ¯B,m

n−1 

n 

w jx j =

j=1

FQB,m (X ) − FPB,m (X ) =

m 1 1 1  1 m (m + 1 ) =1− · i=1− · = , m+1 m m (m + 1 ) 2 2

QB,m

n 

we suppose that x1 ≥ x2 ≥ · · · ≥

 

Q

 

 i i Bim (x ) ≥ P Bim (x ), x ∈ [0, 1]. m m m

i=1

It is clear from Eq. (15) that QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1].  q

q

q

p

p

p

Theorem 3. Let Sq = (s1 , s2 , . . . , sm )T and S p = (s1 , s2 , . . . , sm )T be q p two attitudinal weighting vectors where si , si (i = 1, 2, . . . , m)  q are given by Eq. (9). For Sq and S p , let Q ( mi ) = ik=1 sk and  p q p p q P ( mi ) = ik=1 sk (i = 1, 2, . . . , m). If sk sl ≥ sk sl for any l ≥ k (k, l = 1, 2, . . . , m), then Q ( mi ) ≥ P ( mi ) for all i = 1, 2, . . . , m.

Proof. Similar to Theorem 2 in Liu and Han (2008). Omitted.

(sq1 , sq2 , . . . , sqm )T

Theorem 4. Let Sq = two attitudinal weighting vectors



p p p and S p = (s1 , s2 , . . . , sm )T be q p where si , si (i = 1, 2, . . . , m)

K. Guo / European Journal of Operational Research 254 (2016) 226–235

231

Table 1 Specifications of the alternative prototype missiles.

Fig. 1. Consistency of the OWA aggregations guided by QB,m and PB,m .

 q are given by Eq. (9). For Sq and S p , let Q ( mi ) = ik=1 sk and i p q q p p i P ( m ) = k=1 sk (i = 1, 2, . . . , m). If s j − s j+1 ≥ s j − s j+1 for any j = 1, 2, . . . , m − 1, then

Q ( mi )

≥

P ( mi )

for all i = 1, 2, . . . , m.

Proof. Similar to Theorem 3 in Liu and Han (2008). Omitted.



Theorem 5. Let QB,m and PB,m be two SEVSI-induced quantifiers with Bernstein polynomials. For any argument vectorX = (x1 , x2 , . . . , xn )T , FQB,m (X ) ≥ FPB,m (X )if and only if QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1]. Proof: 1. ) Sufficiency. We shall prove “QB,m (x ) ≥ PB,m (x ) ⇒ FQB,m (X ) ≥ FPB,m (X )”. Given the fact that QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1] and x j ≥ x j+1 for all j = 1, 2, . . . , n − 1, it is clear from Eq. (17) that FQB,m (X ) ≥ FPB,m (X ). 2) Necessity. We shall prove “FQB,m (X ) ≥ FPB,m (X ) ⇒ QB,m (x ) ≥ PB,m (x )”. It is obvious that QB,m (0 ) = PB,m (0 ) = 0 and QB,m (1 ) = PB,m (1 ) = 1. In view of the arbitrariness of X, let X (k ) = (1, 1, . . . , 1, 0, 0, . . . , 0)T (k = 1, 2, . . . , n − 1). Since FQB,m (X (k) ) ≥



 k

   

FPB,m (X (k ) )

QB,m ( nk )

n−k

for

all

k = 1, 2, . . . , n − 1,

then,

from

Eq.

≥ for all k = 1, 2, . . . , n − 1. Given the arbitrary parameters k and n, it is clear that QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1].  Theorem 6. Let QB,m and PB,m be two SEVSI-induced quantifiers with Bernstein polynomials. Let λQB,m and λPB,m be two attitudinal characters associated with QB,m and PB,m , respectively. If QB,m (x ) ≥ PB,m (x ) for any x ∈ [0, 1], then λQB,m ≥ λPB,m . then

Corollary. Let QB,m and PB,m be two SEVSI-induced quantifiers with Bernstein polynomials. Let λQB,m and λPB,m be two attitudinal characters associated with QB,m and PB,m , respectively. For any argument vector X = (x1 , x2 , . . . , xn )T , if FQB,m (X ) ≥ FPB,m (X ) then λQB,m ≥ λPB,m . Proof. Trivial from Theorem 5 and Theorem 6.

Accuracy/km (C1 )

Payload/kg (C2 )

Mobility/km/h (C3 )

Price/106 $ (C4 )

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

2.0 2.5 4.8 2.2 3.0 2.2 2.4 2.8 2.6 3.5

500 540 480 520 580 510 530 560 550 520

[55,56] [30,40] [25,30] [35,40] [30,35] [40,45] [33,40] [28,32] [36,40] [48,50]

[4.7,5.7] [4.2,5.2] [5.0,6.0] [4.5,5.5] [4.8,5.5] [4.8,5.1] [4.0,4.5] [4.8,5.3] [4.5,5.0] [4.8,5.8]

easier to handle compared with a piecewise function with many pieces. Thus this study, as a direct continuation of our previous work, actually develops a SEVSI-induced quantifier in a relatively simple functional form with the good theoretical properties and the pleasing operability for practical use.

4. Illustrative examples This section aims at illustrating and examining, by two numerical examples (adapted from (Guo, 2014)), the SEVSI-induced quantifier with Bernstein polynomials through a comparative analysis of the quantifier with the piecewise linear interpolation as presented in our previous work. Example 1. Quantifier generation. Assume there is no available quantifier for an involved DM who may show, let’s say, a consistent preference or attitude towards attribute values in assessment. We then decide to generate a personalized one for him/her by our developed technique.

(17),

PB,m ( nk )

for any x ∈ [0, 1], Proof. Since Q (x ) ≥ PB,m (x ) 1  1B,m 0 QB,m (x )dx ≥ 0 PB,m (x )dx, i.e., λQB,m ≥ λPB,m . 

Types

Step 1. Prepare a collection of multi-attribute sample information involving the assessments for ten prototype missiles, the specifications of which are shown in Table 1. Obviously, the attribute Price (C4 ) is a cost criterion and the others are benefit ones. This set of sample information can be regarded as a multi-attribute decision matrix with 10 alternatives and 4 criteria, denoted by D = (di j [d˜i j ] )10×4 . Step 2. Let’s work out the normalization of D first. For the crisp values di j in D, let d¯i j be the normalization of di j , where

d¯i j =



Generally speaking, the fact of QB,m (x ) ≥ PB,m (x ) or FQB,m (X ) ≥ FPB,m (X ) cannot be deduced simply from λQB,m ≥ λPB,m . The consistency of the OWA aggregations guided by QB,m and PB,m is roughly summarized in Fig. 1 for a better understanding of this part. It is clear from above that our personalized quantifier shown as Eq. (15) not only is denoted by a BUM function, but also inherits many excellent properties of Bernstein polynomials that may help enhance the performance of global approximation of the fitting functions. In fact, as the value of m increases, the fitting functions are achieving better approximation performance. More notably, this functional form, a polynomial of higher degree, is much

⎧ d ⎪ ⎪  i j ⎪ ⎪ 10 ⎪ 2 ⎨ k=1 dk j

| j ∈ J+

1/di j ⎪ ⎪ ⎪  ⎪  ⎪ 2 10 ⎩ k=1 (1/dk j )

| j ∈ J−

,

i = 1, 2, . . . , 10, j = 1, 2

(18) while for the intervals d˜i j = [d˜iLj , d˜U ], let dˆi j = [dˆiLj , dˆU ] be the norij ij malization of d˜i j , where

dˆiLj =

⎧ d˜iLj ⎪ ⎪ ⎪  ⎪ ⎪ 10  ˜U 2 ⎪ ⎨ k=1 dk j

| j ∈ J+

⎪ 1/d˜Ui j ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ 10 (1/d˜L )2 k=1 kj

| j ∈ J−

,

232

K. Guo / European Journal of Operational Research 254 (2016) 226–235

dˆUi j =

⎧ d˜Ui j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜L 2 ⎨ 10 k=1 (dk j )

| j ∈ J+

⎪ 1/d˜iLj ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ 10 (1/d˜U )2

| j ∈ J−

k=1

− 28.476x5 + 18.06x4 + 2.28x3 + 0.18x2 + 0.02x, x ∈ [0, 1]. ,

i = 1, 2, . . . , 10, j = 3, 4.

λQ (1) =

kj

(19) J−

Here represents a set of benefit criteria, and represents a set of cost ones. It is clear that d¯i j ∈ [0, 1](i = 1, 2, . . . , 10; j = 1, 2) and dˆi j ⊆ [0, 1](i = 1, 2, . . . , 10; j = 3, 4). For simplicity, we take here the inferior limits of the normalized intervals dˆi j (i = 1, 2, . . . , 10; j = 3, 4). After the rearrangement of the values of each criterion in descending order, the corresponding reordered normalized decision matrix is shown as

0.522

⎢0.381 ⎢0.326 ⎢ ⎢0.304 ⎢   ⎢0.283 =⎢ D¯ = d¯i j 10×4 ⎢0.272 ⎢0.261 ⎢ ⎢0.239 ⎣0.239 0.217



B,10

J+

⎡

(1 ) The attitudinal character associated with QB, can then be 10 measured as

0.346 0.334 0.328 0.322 0.316 0.310 0.310 0.304 0.298 0.287

⎤

0.419 0.366 0.305 0.274 0.267 0.252 0.229 0.229 0.213 0.191

0.322 0.290⎥ 0.284⎥ ⎥ 0.278⎥ ⎥ 0.273⎥ ⎥. 0.263⎥ 0.263⎥ ⎥ 0.254⎥ ⎦ 0.250 0.241

Step 3. The involved DM is then asked to provide his/her expected or satisfactory value for each criterion with respect to D¯ = (d¯i j )10×4 . Assume the set of expected values provided by this DM is

ν¯ 1 = (0.29, 0.32, 0.25, 0.27 )T .

0

ν¯ 2 = (0.22, 0.29, 0.21, 0.25 )T , which is showing a preference for lower sample values as each component weight of the vector ν¯ 2 is near the bottom of the corresponding column vector of D¯ . With this vector, we get

S2 = (0.0 01, 0.0 03, 0.0 07, 0.012, 0.019, 0.031, 0.060, 0.167, 0.360, 0.340 )T . Clearly, the distribution of S2 presents a one-sided tendency towards the bottom, implying that this DM is characterized by a pessimistic attitude represented here as the quantifier Q2 . Similarly,

Q2

1

= 0.001, Q2

Q2

6

10

9

10

= 0.073, Q2

Q1

10 10

= 0.002, Q1 = 0.194, Q1

6

2

10

5

10

= 0.008, Q1

3

10

7

= 0.037,

8

(1 ) QB, (x ) = 10

i=1



Q1

8

10

= 0.011,

= 0.300,

(2 ) QB, (x ) = 10

10  i=1



Q2



i 10−i C i xi ( 1 − x ) 10 10

= 0.366x10 − 2.01x9 + 1.17x8 + 0.84x7 + 0.504x5 − 0.21x4 + 0.24x3 + 0.09x2 (2 ) The attitudinal character associated with QB, can then be 10 measured as

λQ (2) =



1 0

(2 ) QB, (x )dx = 0.204, 10

which is showing a lower level of the pessimistic attitude Q2 . On the other hand, if the DM provides his/her set of expected values as

ν¯ 3 = (0.50, 0.34, 0.40, 0.30 )T , = 0.962,

Step 5. On the basis of these discrete values of Q1 , fit a contin(1 ) uous quantifier QB, by using Eq. (15), shown as 10 10 

10

= 0.133, Q2

10

(2 ) A continuous quantifier QB, can then be generated as 10

B,10

= 0.557,

= 0.833, Q1 = 0.925, Q1 10 10 10 9

Q1 = 0.989, Q1 (1 ) = 1. 10 Q1

7

3

+ 0.01x, x ∈ [0, 1].

It is clear from S1 that the significant weights are evenly dispersed around the middle positions, implying that this DM is characterized by a neutral attitude represented here as the quantifier  Q1 . According to Q1 ( mi ) = ik=1 sk for all i = 1, 2, . . . , 10, we get

4

= 0.004, Q2

= 0.660, Q2 (1 ) = 1.

0.027, 0.011 )T .

Q1

2

10 10 4

5

Q2 = 0.023, Q2 = 0.042, 10 10

S1 = (0.0 02, 0.0 06, 0.029, 0.157, 0.363, 0.276, 0.092, 0.037,

1

(1 ) QB, (x )dx = 0.501, 10

which is showing an actual level of the neutral attitude Q1 . (1 ) Step 6. The OWA aggregation under the guidance of QB, 10 can then be implemented by using Eq. (3) for this DM’s future decisions.  Let’s look at some other cases for the different subjective expected values of D¯ = (d¯i j )10×4 . Assume the DM provides his/her set of expected values as

Q2

Given the middle position where each component weight of the vector ν¯ 1 is nearly located in the corresponding column vector of D¯ , it is clear to see that this DM shows a preference for the sample values in the middle between the large and small ones. This is clearly a neutral attitude. Step 4. With the observed data D¯ and ν¯ 1 , determine the attitudinal weighting vector associated directly with this DM by using Eq. (9), denoted by

1



i 10−i C i xi ( 1 − x ) 10 10

= −5.394x10 + 42.35x9 − 116.55x8 + 137.04x7 − 48.51x6

which is an indication of preference for higher sample values as each component weight of the vector ν¯ 3 is at the top of the corresponding column vector of D¯ , we then have

S3 = (0.809, 0.071, 0.028, 0.020, 0.017, 0.014, 0.012, 0.011, 0.010, 0.008 )T . Clearly, the distribution of S3 presents a one-sided tendency towards the top, implying that this DM is characterized by an optimistic attitude represented here as the quantifier Q3 . In this case,

Q3

1

10

= 0.809, Q3

2

10

= 0.880, Q3

3

10

= 0.908,

K. Guo / European Journal of Operational Research 254 (2016) 226–235

233

where si (i = 1, 2, . . . , m) are given by Eq. (9). More specifically, the quantifiers generated from the above Sk (k = 1, 2, 3) are respectively represented as

QD¯ ,ν¯ 1 (x ) =

QD¯ ,ν¯ 2 (x ) =

Fig. 2. Geometrical representation of the exemplified quantifiers with Bernstein polynomials

Q3

4

10

6

= 0.928, Q3

5

10

7

= 0.945,

8

= 0.959, Q3 = 0.971, Q3 10 10 10 9

Q3 = 0.992, Q3 (1 ) = 1. 10 Q3

QD¯ ,ν¯ 3 (x ) = = 0.982,

A continuous quantifier QB,10 can then be generated as (3 ) QB, (x ) = 10



Q3

i=1

= −0.57x

i 10−i C i xi ( 1 − x ) 10 10 9

1

7

+ 5.69x − 25.83x + 70.32x − 127.05x

6

+ 8.09x, x ∈ [0, 1]. (3 ) The attitudinal character associated with QB, is measured as 10

B,10



1 0

(3 ) QB, (x )dx = 0.851, 10

which is showing a higher level of the optimistic attitude Q3 . Geometrical representation of the exemplified quantifiers with Bernstein polynomials is shown in Fig. 2 in an effort to make these quantifiers quite understood.  For a comparative analysis, let’s look at another functional form of our personalized quantifier with the piecewise linear interpolation as presented in our previous work. Under the circumstances, this type of SEVSI-induced quantifier is defined by Guo (2014)

QD¯ ,ν¯ (x ) = (mx − i )si +

i  k=1

sk ,

i i−1 ≤x≤ , m m

i = 1, 2, . . . , m, (20)

and the attitudinal character associated with QD¯ ,ν¯ is measured by Guo (2014)

λQD¯ ,ν¯ = AC (QD¯ ,ν¯ ) =

 0

1

0.1668x + 0.8600

0.1448x + 0.8710 ⎪ ⎪ ⎪ 0.1244x + 0.8833 ⎪ ⎪ ⎪ ⎪ 0.1094x + 0.8937 ⎪ ⎪ ⎪ ⎪ ⎩0.1027x + 0.8991

0.0 ≤ x ≤ 0.1 0.1 ≤ x ≤ 0.2 0.2 ≤ x ≤ 0.3 0.3 ≤ x ≤ 0.4 0.4 ≤ x ≤ 0.5 . 0.5 ≤ x ≤ 0.6 0.6 ≤ x ≤ 0.7 0.7 ≤ x ≤ 0.8 0.8 ≤ x ≤ 0.9 0.9 ≤ x ≤ 1.0 0.0 ≤ x ≤ 0.1 0.1 ≤ x ≤ 0.2 0.2 ≤ x ≤ 0.3 0.3 ≤ x ≤ 0.4 0.4 ≤ x ≤ 0.5 . 0.5 ≤ x ≤ 0.6 0.6 ≤ x ≤ 0.7 0.7 ≤ x ≤ 0.8 0.8 ≤ x ≤ 0.9 0.9 ≤ x ≤ 1.0

λQD¯ ,ν¯ = 0.501, λQD¯ ,ν¯ = 0.175, λQD¯ ,ν¯ = 0.886. 8

+ 158.76x5 − 138.6x4 + 83.4x3 − 33.21x2

λQ (3) =

0.1856x − 0.0514

0.3104x − 0.1137 ⎪ ⎪ ⎪ 0.5989x − 0.2868 ⎪ ⎪ ⎪ ⎪ 1.6741x − 1.0395 ⎪ ⎪ ⎪ ⎪ ⎩3.6028x − 2.5824 3.3992x − 2.3992 ⎧ 8.0761x ⎪ ⎪ ⎪0.7134x + 0.7363 ⎪ ⎪ ⎪ ⎪ 0.2777x + 0.8234 ⎪ ⎪ ⎪ ⎪0.2001x + 0.8467 ⎨

0.0 ≤ x ≤ 0.1 0.1 ≤ x ≤ 0.2 0.2 ≤ x ≤ 0.3 0.3 ≤ x ≤ 0.4 0.4 ≤ x ≤ 0.5 , 0.5 ≤ x ≤ 0.6 0.6 ≤ x ≤ 0.7 0.7 ≤ x ≤ 0.8 0.8 ≤ x ≤ 0.9 0.9 ≤ x ≤ 1.0

The attitudinal characters associated with these quantifiers can then be measured as



10

3.6268x − 1.2578

2.7600x − 0.8244 ⎪ ⎪ ⎪ 0.9200x + 0.2796 ⎪ ⎪ ⎪ ⎪ 0.3704x + 0.6643 ⎪ ⎪ ⎪ ⎪ ⎩0.2712x + 0.7436 0.1226x + 0.8774 ⎧ 0.0109x ⎪ ⎪ ⎪ 0.0291x − 0.0018 ⎪ ⎪ ⎪ ⎪ 0.0684x − 0.0097 ⎪ ⎪ ⎪ ⎪0.1206x − 0.0254 ⎨

0.0848x + 0.9152

(3 )

10 

⎧ 0.0154x ⎪ ⎪0.0589x − 0.0044 ⎪ ⎪ ⎪ ⎪ ⎪ 0.2874x − 0.0500 ⎪ ⎪ ⎪ ⎪ ⎨1.5673x − 0.4340

QD¯ ,ν¯ (x )dx = 1 −

m 1  1 isi + , m 2m i=1

3

Geometrical representation of these quantifiers with the piecewise linear interpolation is shown in Fig. 3.  It is clear from above that the SEVSI-induced quantifier with the piecewise linear interpolation may sometimes suffer from some limitations mainly concerning the treatment of too many pieces, the number of which depends on the number of the alternatives in the given sample. In fact, a piecewise function with a large number of pieces may bring some confusion to users and become difficult to handle. Generally, the use of piecewise functions with pieces more than 8 is rarely considered for practical applications (Cheney, 1982). Besides, there is a need, if necessary, for significant improvement on the smoothness of the fitted curves by this interpolation. By contrast, the quantifier with Bernstein polynomials of higher degree can eliminate these limitations with better theoretical properties such as the excellent shape-preserving property and a sufficient degree of smoothness. More notably, the quantifier in this functional form has become much easier to handle either for theoretical analysis or for practical applications, as mentioned before. Example 2. Quantifier examination. Assume there is a multiattribute decision matrix with 4 alternatives ai (i = 1, 2, 3, 4) and 3 benefit criteria, shown as

⎡



(21)

2

P = pi j

 4×3

0.691 ⎢0.739 =⎣ 0.672 0.711

0.713 0.730 0.705 0.722

⎤

0.756 0.279⎥ . 0.733⎦ 0.325

234

K. Guo / European Journal of Operational Research 254 (2016) 226–235

Table 2 Evaluations of alternatives under the guidance of the quantifiers with Bernstein polynomials Quantifiers (1 )

QB,10 (2 ) QB, 10 (3 ) QB, 10

Attitudinal characters

OWA weighting vectors

λ = 0.501 λ(2) = 0.204 λ(3) = 0.851

W = (0.197,0.612,0.191) W(2) = (0.022,0.169,0.809)T W(3) = (0.892,0.073,0.035)T

(1)

(1)

Quantifier guided OWA aggregations T

Ranking of the alternatives

F (a1 ) = 0.717, F (a2 ) = 0.646, F (a3 ) = 0.704, F (a4 ) = 0.639. F(2) (a1 ) = 0.696, F(2) (a2 ) = 0.366, F(2) (a3 ) = 0.679, F(2) (a4 ) = 0.399. F(3) (a1 ) = 0.751, F(3) (a2 ) = 0.722, F(3) (a3 ) = 0.729, F(3) (a4 ) = 0.707. (1)

(1)

(1)

(1)

a1 a3 a2 a4 a1 a3 a4 a2 a1 a3 a2 a4

Table 3 Evaluations of alternatives under the guidance of the quantifiers with the piecewise linear interpolation. Quantifiers

Attitudinal characters

OWA weighting vectors

Quantifier guided OWA aggregations

Ranking of the alternatives

QD¯ ,ν¯ 1 QD¯ ,ν¯ 2 QD¯ ,ν¯ 3

λ1 = 0.501 λ2 = 0.175 λ3 = 0.886

W1 = (0.088,0.805,0.107)T W2 = (0.015,0.098,0.887)T W3 = (0.913,0.053,0.034)T

F1 (a1 ) = 0.714, F1 (a2 ) = 0.683, F1 (a3 ) = 0.704, F1 (a4 ) = 0.671. F2 (a1 ) = 0.694, F2 (a2 ) = 0.330, F2 (a3 ) = 0.676, F2 (a4 ) = 0.369. F3 (a1 ) = 0.752, F3 (a2 ) = 0.723, F3 (a3 ) = 0.729, F3 (a4 ) = 0.708.

a1 a3 a2 a4 a1 a3 a4 a2 a1 a3 a2 a4

points to the necessity of considering the different attitudes of the involved DMs in increasingly complex environments. Let’s now look at the evaluations of ai (i = 1, 2, 3, 4) with the OWA aggregation under the guidance of QD¯ ,ν¯ (k = 1, 2, 3) so as to k

make a direct comparison of results between these two kinds of functional form. More details of these are summarized and shown in Table 3, where λk , Wk , and Fk (· )(k = 1, 2, 3) are determined by Eqs. (21), (2), and (3), respectively. By comparison with Tables 2 and 3, we clearly see that our personalized quantifier, either represented as Bernstein polynomials of higher degree or as piecewise linear functions, can surely produce the consistent results reflecting the decision attitudes or behavior intentions of the involved DMs in terms of what we expect. Thus our work on this research topic has been proven fruitful. By contrast, the quantifier with Bernstein polynomials of higher degree, characterized by the excellent theoretical properties and the pleasing operability, definitely outperforms the one with the piecewise linear interpolation in many aspects of geometrical characteristics and operability, and therefore it could be considered to put to practical use in many areas. 5. Conclusions

Fig. 3. Geometrical representation of the exemplified quantifiers with the piecewise linear interpolation

For convenience, let

a1 = (0.691, 0.713, 0.756 )T ,

a2 = (0.739, 0.730, 0.279 )T ,

a3 = (0.672, 0.705, 0.733 ) ,

a4 = (0.711, 0.722, 0.325 )T .

T

Let’s investigate the preliminary evaluations of ai (i = 1, 2, 3, 4) (k ) with the OWA aggregation under the guidance of QB, (k = 1, 2, 3) 10 generated from Example 1. More details are provided as shown in Table 2, where λ(k ) , W (k ) , and F (k ) (· )(k = 1, 2, 3) are determined by Eqs. (16), (2), and (3), respectively. It is clear from Table 2 that in terms of the idea of OWA aggregation, the OWA weighting vector W (1 ) is an indication of preference for argument values in the middle between the large and small ones as the significant weights are dispersed as evenly as possible around the middle positions. This is clearly a reflection of neutral attitude. Similarly, the vector W (2 ) is an indication of preference for lower argument values as more of the total weight moves to the weights at the bottom. This is clearly a reflection of pessimistic attitude. And the vector W (3 ) is an indication of preference for higher argument values as more of the total weight moves to the top. This is clearly a reflection of optimistic attitude. Also, from Table 2 it can be seen that for the same decision information, different attitudes may lead to different assessments, which

In this paper, we develop a new type of SEVSI-induced quantifier with Bernstein polynomials of higher degree, thus further perfecting this kind of personalized quantifier as proposed in our previous work. The developed quantifier actually inherits many excellent properties of Bernstein polynomials, and has become much easier to handle in comparison with the one with the piecewise linear interpolation whether for theoretical analysis or for practical applications. As for the application of the SEVSI-induced quantifier in different functional forms, here is some advice from a more practical point of view: if there is only a small quantity of the alternatives in the given sample (more specifically, the number of the alternatives is less than 8) and there is no need for the smoothness of the fitted curves, the quantifier with the piecewise linear interpolation would be a good choice given its simplicity and relatively fast convergence rate in the circumstances; otherwise, the quantifier with Bernstein polynomials should be taken into account. Just because of what we did for the quantifier, it is becoming much more efficient and flexible to deal with such a complex case in which the involved DMs are allowed to have some unusual even extreme decision attitudes. Also, it may help to predict the involved DMs’ future decisions, and with good reason. Acknowledgments This work is partially supported by the Ph.D. Research Startup Foundation of Liaoning University. The author would like to thank

K. Guo / European Journal of Operational Research 254 (2016) 226–235

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