# A Study of SUOWA Operators in Two Dimensions

Authors: Bonifacio Llamazares
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

SUOWA operators are a new class of aggregation functions that simultaneously generalize weighted means and OWA operators. They are Choquet integral-based operators with respect to normalized capacities; therefore, they possess some interesting properties such as continuity, monotonicity, idempotency, compensativeness, and homogeneity of degree 1. In this paper, we focus on two dimensions and show that any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator.


---

## Body

## 1. Introduction

The study of aggregation operators has received special attention in the last years. This is due to the extensive applications of these functions for aggregating information in a wide variety of areas. Two of the best-known aggregation operators are the weighted means and the ordered weighted averaging (OWA) operators (Yager [1]). Both classes of functions are defined by means of weighting vectors, but their behavior is quite different. Weighted means allow weighting each information source in relation to their reliability while OWA operators allow weighting the values according to their ordering.Although both families of operators allow solving a wide range of problems, both weightings are necessary in some contexts. Some examples of these situations have been given by several authors (see, for instance, Torra [2–4], Torra and Godo [5, pages 160-161], Torra and Narukawa [6, pages 150-151], Roy [7], Yager and Alajlan [8], and Llamazares [9] and the references therein) in fields as diverse as robotics, vision, fuzzy logic controllers, constraint satisfaction problems, scheduling, multicriteria aggregation problems, and decision-making.A typical situation where both weightings are necessary is the following (Llamazares [9]): suppose we have several sensors to measure a physical property. On the one hand, sensors may be of different quality and precision, so a weighted mean type aggregation is necessary. On the other hand, to prevent a faulty sensor from altering the measurement, we might consider an OWA type aggregation where the maximum and minimum values are not taken into account. A similar situation occurs when a committee of experts has to assess several candidates or proposals. On the one hand, a weighted mean type aggregation is suitable for reflecting the expertness or the confidence in the judgment of each expert. On the other hand, an OWA type aggregation allows us to deal with situations where an expert feels excessive acceptance or rejection towards some of the candidates or proposals.Different aggregation operators have appeared in the literature to deal with this kind of problems. A usual approach is to consider families of functions parameterized by two weighting vectors, one for the weighted mean and the other one for the OWA type aggregation, which generalize weighted means and OWA operators in the following sense. A weighted mean (or an OWA operator) is obtained when the other weighting vector has a “neutral” behavior; that is, it is(1/n,…,1/n) (see Llamazares [10] for an analysis of some functions that generalize the weighted means and the OWA operators in this sense). Two of the solutions having better properties are the weighted OWA (WOWA) operator, proposed by Torra [3], and the semiuninorm based ordered weighted averaging (SUOWA) operator, introduced by Llamazares [9].The good properties of WOWA and SUOWA operators are due to the fact that they are Choquet integral-based operators with respect to normalized capacities. In the case of SUOWA operators, their capacities are the monotonic cover of certain games, which are defined by using the capacities associated with the weighted means and the OWA operators and “assembling” these values through semiuninorms with neutral element1/n.Because of their good properties, it seems interesting to analyze the behavior of SUOWA operators from different points of view. In this paper, we consider the case of two dimensions that, although simple, is attractive from a theoretical point of view, and we show that any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator.The remainder of the paper is organized as follows. In Section2 we recall the concepts of semiuninorm and uninorm and give some interesting examples of such functions. Section 3 is devoted to Choquet integral, including some of the most important particular cases: weighted means, OWA operators, and SUOWA operators. In Section 4, we give the main results of the paper. Finally, some concluding remarks are provided in Section 5.


## 2. Semiuninorms and Uninorms

Throughout the paper, we will use the following notation:N={1,…,n}; given A⊆N, A denotes the cardinality of A; vectors are denoted in bold and η denotes the tuple (1/n,…,1/n)∈Rn. We write x≥y if xi≥yi for all i∈N. For a vector x∈Rn, [·] and (·) denote permutations such that x[1]≥⋯≥x[n] and x(1)≤⋯≤x(n).Semiuninorms are a class of necessary functions in the definition of SUOWA operators. They are monotonic and have a neutral element in the interval[0,1]. These functions were introduced by Liu [11] as a generalization of uninorms, which, in turn, were proposed by Yager and Rybalov [12] as a generalization of t-norms and t-conorms.Before introducing the concepts of semiuninorm and uninorm, we recall some well-known properties of aggregation functions.Definition 1.
LetF:Rn→R be a function.(1)
F is symmetric if F(xσ(1),…,xσ(n))=F(x1,…,xn) for all x∈Rn and for all permutation σ of N.(2)
F is monotonic if x≥y implies F(x)≥F(y) for all x,y∈Rn.(3)
F is idempotent if F(x,…,x)=x for all x∈R.(4)
F is compensative (or internal) if min(x)≤F(x)≤max(x) for all x∈Rn.(5)
F is homogeneous of degree 1 (or ratio scale invariant) if F(rx)=rF(x) for all x∈Rn and for all r>0.Definition 2.
LetU:[0,1]2→[0,1].(1)
U is a semiuninorm if it is monotonic and possesses a neutral element e∈[0,1] (U(e,x)=U(x,e)=x for all x∈[0,1]).(2)
U is a uninorm if it is a symmetric and associative (U(x,U(y,z))=U(U(x,y),z) for all x,y,z∈[0,1]) semiuninorm.We denote byUe (resp., Uie) the set of semiuninorms (resp., idempotent semiuninorms) with neutral element e∈[0,1].SUOWA operators are defined by using semiuninorms with neutral element1/n. Moreover, they have to belong to the following subset (see Llamazares [9]):(1)U~1/n=U∈U1/n∣U1k,1k≤1k∀k∈N.Obviously,Ui1/n⊆U~1/n. Notice that the smallest and the largest elements of U~1/n are, respectively, the following semiuninorms: (2)U⊥x,y=max⁡x,yifx,y∈1n,12,0ifx,y∈0,1n2,min⁡x,yotherwise,U⊤x,y=1kifx,y∈Ik∖Ik+1,  whereIk=1n,1k2k∈N∖n,min⁡x,yifx,y∈0,1n2,max⁡x,yotherwise.In the case of idempotent semiuninorms, the smallest and the largest elements ofUi1/n are, respectively, the following uninorms (which were given by Yager and Rybalov [12]):(3)Uminx,y=max⁡x,yifx,y∈1n,12,min⁡x,yotherwise,Umaxx,y=min⁡x,yifx,y∈0,1n2,max⁡x,yotherwise.In addition to the previous ones, several procedures to construct semiuninorms have been introduced by Llamazares [13]. One of them, which is based on ordinal sums of aggregation operators, allows us to get continuous semiuninorms. Some of the most relevant continuous semiuninorms obtained are the following:(4)UTLx,y=max⁡x,yifx,y∈1n,12,max⁡x+y-1n,0otherwise,UP~x,y=max⁡x,yifx,y∈1n,12,nxyotherwise,UTMx,y=max⁡x,yifx,y∈1n,12,min⁡x,yifx,y∈0,1n2,x+y-1notherwise,UPx,y=max⁡x,yifx,y∈1n,12,min⁡x,yifx,y∈0,1n2,nxyotherwise.Notice that the last two semiuninorms are also idempotent. The plots of all these semiuninorms are given, for the casen=4, in Figures 1–8.Figure 1
The semiuninormU⊥ for n=4.Figure 2
The semiuninormU⊤ for n=4.Figure 3
The uninormUmin for n=4.Figure 4
The uninormUmax for n=4.Figure 5
The uninormUTL for n=4.Figure 6
The uninormUP~ for n=4.Figure 7
The uninormUTM for n=4.Figure 8
The uninormUP for n=4.


## 3. Choquet Integral

The notion of Choquet integral is based on that of capacity (see Choquet [14] and Murofushi and Sugeno [15]). The concept of capacity resembles that of probability measure but in the definition of the former additivity is replaced by monotonicity (see also fuzzy measures in Sugeno [16]). A game is then a generalization of a capacity where the monotonicity is no longer required.Definition 3.
(1) A gameυ on N is a set function, υ:2N→R satisfying υ(⌀)=0.
(2) A capacity (or fuzzy measure)μ on N is a game on N satisfying μ(A)≤μ(B) whenever A⊆B. In particular, it follows that μ:2N→[0,∞). A capacity μ is said to be normalized if μ(N)=1.A straightforward way to get a capacity from a game is to consider the monotonic cover of the game (see Maschler and Peleg [17] and Maschler et al. [18]).Definition 4.
Letυ be a game on N. The monotonic cover of υ is the set function υ^ given by (5)υ^A=maxB⊆A⁡υB.Some basic properties ofυ^ are given in the sequel.Remark 5.
Letυ be a game on N. Then, one has the following:(1)
υ^ is a capacity.(2)
Ifυ is a capacity, then υ^=υ.(3)
Ifυ(A)≤1 for all A⊆N and υ(N)=1, then υ^ is a normalized capacity.Although the Choquet integral is usually defined as a functional (see, for instance, Choquet [14], Murofushi and Sugeno [15], and Denneberg [19]), in this paper we consider the Choquet integral as an aggregation function over Rn (see, for instance, Grabisch et al. [20, page 181]). Moreover, we define the Choquet integral for all vectors of Rn instead of nonnegative vectors given that we are actually considering the asymmetric Choquet integral with respect to μ (on this, see again Grabisch et al. [20, page 182]).Definition 6.
Letμ be a capacity on N. The Choquet integral with respect to μ is the function Cμ:Rn→R given by(6)Cμx=∑i=1nμBixi-xi-1,where B(i)={(i),…,(n)}, and one uses the convention x(0)=0.It is worth noting that the Choquet integral has several properties which are useful in certain information aggregation contexts (see, for instance, Grabisch et al. [20, pages 192-193 and page 196]).Remark 7.
Letμ be a capacity on N. Then, Cμ is continuous, monotonic, and homogeneous of degree 1. Moreover, it is idempotent and compensative when μ is a normalized capacity.Notice that the Choquet integral can also be represented by using decreasing sequences of values (see, for instance, Torra [21] and Llamazares [9]):(7)Cμx=∑i=1nμAixi-xi+1,where A[i]={[1],…,[i]}, and we use the convention x[n+1]=0.From the previous expression, it is straightforward to show explicitly the weights of the valuesx[i] by representing the Choquet integral as follows: (8)Cμx=∑i=1nμAi-μAi-1xi,where we use the convention A[0]=⌀.


### 3.1. Weighted Means and OWA Operators

Weighted means and OWA operators (Yager [1]) are well-known functions in the field of aggregation operators. Both families of functions are defined in terms of weight distributions that add up to 1.Definition 8.
A vectorq∈Rn is a weighting vector if q∈[0,1]n and ∑i=1n‍qi=1.The set of all weighting vectors ofRn will be denoted by Wn.Definition 9.
Letp be a weighting vector. The weighted mean associated with p is the function Mp:Rn→R given by(9)Mpx=∑i=1npixi.Definition 10.
Letw be a weighting vector. The OWA operator associated with w is the function Ow:Rn→R given by(10)Owx=∑i=1nwixi.It is well known that weighted means and OWA operators are a special type of Choquet integral (see, for instance, Fodor et al. [22], Grabisch [23, 24], or Llamazares [9]).Remark 11.
(1) Ifp is a weighting vector, then the weighted mean Mp is the Choquet integral with respect to the normalized capacity μp(A)=∑i∈Api.
(2) Ifw is a weighting vector, then the OWA operator Ow is the Choquet integral with respect to the normalized capacity μw(A)=∑i=1Awi.So, according to Remark7, weighted means and OWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1. Moreover, in the case of OWA operators, given that the values of the variables are previously ordered in a decreasing way, they are also symmetric.


### 3.2.  SUOWA Operators

SUOWA operators were introduced by Llamazares [9] in order to consider situations where both the importance of information sources and the importance of values had to be taken into account. These functions are Choquet integral-based operators where their capacities are the monotonic cover of certain games. These games are defined by using semiuninorms with neutral element 1/n and the values of the capacities associated with the weighted means and the OWA operators. To be specific, the games from which SUOWA operators are built are defined as follows.Definition 12.
Letp and w be two weighting vectors and let U∈U~1/n. (1)
The game associated withp, w, and U is the set function υp,wU:2N→R defined by(11)υp,wUA=AUμpAA,μwAA=AU∑i∈ApiA,∑i=1AwiAifA≠⌀ and υp,wU(⌀)=0.(2)
υ^p,wU, the monotonic cover of the game υp,wU, will be called the capacity associated with p, w, and U.Notice thatυp,wU(N)=1. Moreover, since U∈U~1/n, we have υp,wU(A)≤1 for all A⊆N (see Llamazares [9]). Therefore, according to the third item of Remark 5, υ^p,wU is always a normalized capacity.Definition 13.
Letp and w be two weighting vectors and let U∈U~1/n. The SUOWA operator associated with p, w, and U is the function Sp,wU:Rn→R given by(12)Sp,wUx=∑i=1nsixi,where si=υ^p,wU(A[i])-υ^p,wU(A[i-1]) for all i∈N, υ^p,wU is the capacity associated with p, w, and U, and A[i]={[1],…,[i]} (with the convention that A[0]=⌀).According to expression (7), the SUOWA operator associated with p, w, and U can also be written as (13)Sp,wUx=∑i=1nυ^p,wUAixi-xi+1.By the choice ofυ^p,wU, we have Sp,ηU=Mp and Sη,wU=Ow for any U∈U~1/n. Moreover, by Remark 7 and given that υ^p,wU is a normalized capacity, SUOWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1.


## 3.1. Weighted Means and OWA Operators

Weighted means and OWA operators (Yager [1]) are well-known functions in the field of aggregation operators. Both families of functions are defined in terms of weight distributions that add up to 1.Definition 8.
A vectorq∈Rn is a weighting vector if q∈[0,1]n and ∑i=1n‍qi=1.The set of all weighting vectors ofRn will be denoted by Wn.Definition 9.
Letp be a weighting vector. The weighted mean associated with p is the function Mp:Rn→R given by(9)Mpx=∑i=1npixi.Definition 10.
Letw be a weighting vector. The OWA operator associated with w is the function Ow:Rn→R given by(10)Owx=∑i=1nwixi.It is well known that weighted means and OWA operators are a special type of Choquet integral (see, for instance, Fodor et al. [22], Grabisch [23, 24], or Llamazares [9]).Remark 11.
(1) Ifp is a weighting vector, then the weighted mean Mp is the Choquet integral with respect to the normalized capacity μp(A)=∑i∈Api.
(2) Ifw is a weighting vector, then the OWA operator Ow is the Choquet integral with respect to the normalized capacity μw(A)=∑i=1Awi.So, according to Remark7, weighted means and OWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1. Moreover, in the case of OWA operators, given that the values of the variables are previously ordered in a decreasing way, they are also symmetric.


## 3.2.  SUOWA Operators

SUOWA operators were introduced by Llamazares [9] in order to consider situations where both the importance of information sources and the importance of values had to be taken into account. These functions are Choquet integral-based operators where their capacities are the monotonic cover of certain games. These games are defined by using semiuninorms with neutral element 1/n and the values of the capacities associated with the weighted means and the OWA operators. To be specific, the games from which SUOWA operators are built are defined as follows.Definition 12.
Letp and w be two weighting vectors and let U∈U~1/n. (1)
The game associated withp, w, and U is the set function υp,wU:2N→R defined by(11)υp,wUA=AUμpAA,μwAA=AU∑i∈ApiA,∑i=1AwiAifA≠⌀ and υp,wU(⌀)=0.(2)
υ^p,wU, the monotonic cover of the game υp,wU, will be called the capacity associated with p, w, and U.Notice thatυp,wU(N)=1. Moreover, since U∈U~1/n, we have υp,wU(A)≤1 for all A⊆N (see Llamazares [9]). Therefore, according to the third item of Remark 5, υ^p,wU is always a normalized capacity.Definition 13.
Letp and w be two weighting vectors and let U∈U~1/n. The SUOWA operator associated with p, w, and U is the function Sp,wU:Rn→R given by(12)Sp,wUx=∑i=1nsixi,where si=υ^p,wU(A[i])-υ^p,wU(A[i-1]) for all i∈N, υ^p,wU is the capacity associated with p, w, and U, and A[i]={[1],…,[i]} (with the convention that A[0]=⌀).According to expression (7), the SUOWA operator associated with p, w, and U can also be written as (13)Sp,wUx=∑i=1nυ^p,wUAixi-xi+1.By the choice ofυ^p,wU, we have Sp,ηU=Mp and Sη,wU=Ow for any U∈U~1/n. Moreover, by Remark 7 and given that υ^p,wU is a normalized capacity, SUOWA operators are continuous, monotonic, idempotent, compensative, and homogeneous of degree 1.


## 4. The Results

The use of Choquet integral has become more and more extensive in the last years (see, for instance, Grabisch et al. [25] and Grabisch and Labreuche [26]). Although simple, the case n=2 is interesting from a theoretical point of view. Thus, for instance, Grabisch et al. [20, page 204] show that, in this case, any Choquet integral with respect to a normalized capacity can be written as a convex combination of a minimum, a maximum, and two projections; that is, given a normalized capacity μ, there exists a weighting vector λ belonging to W4 such that(14)Cμx1,x2=λ1min⁡x1,x2+λ2max⁡x1,x2+λ3x1+λ4x2.In our case, we are going to show that any Choquet integral with respect to a normalized capacity can be written as a SUOWA operator. Notice that whenn=2, υp,wU is always a normalized capacity for any weighting vectors p and w and for any semiuninorm U. Therefore, given a normalized capacity μ, we need to prove that there exist weighting vectors p and w and a semiuninorm U such that (15)υp,wU1=Up1,w1=μ1,υp,wU2=Up2,w1=μ2,where we use the notations μ1 and μ2 to denote the values μ({1}) and μ({2}), respectively.Firstly we are going to show that, in the case of the semiuninormsU⊥, U⊤, Umin, and Umax, there exist normalized capacities which cannot be expressed as SUOWA operators. For this, we will use the following lemma.Lemma 14.
IfU∈{U⊥,U⊤,Umin,Umax}, then U(x,y)=0.5 if and only if x=y=0.5.Proof.
LetU∈{U⊥,U⊤,Umin,Umax}. Since 0.5 is the neutral element of U, we have U(0.5,0.5)=0.5.
Conversely, supposeU(x,y)=0.5. In Table 1, where 0.5- stands for a value that belongs to [0,0.5) and 0.5+ stands for a value that belongs to (0.5,1], we show the values taken by the semiuninorms U⊥, Umin, Umax, and U⊤ when (x,y)∈[0,1]2. Therefore, if U(x,y)=0.5, then necessarily x=y=0.5.Table 1
Values taken byU⊥, Umin⁡, Umax⁡, and U⊤.
x
y
U
⊥
(
x
,
y
)
U
min
⁡
(
x
,
y
)
U
max
⁡
(
x
,
y
)
U
⊤
(
x
,
y
)
0
.
5
-
0
.
5
-
0
min
⁡
(
x
,
y
)
min
⁡
(
x
,
y
)
min
⁡
(
x
,
y
)
0
.
5
-
0.5
x
x
x
x
0
.
5
-
0
.
5
+
x
x
y
y
0.5
0
.
5
-
y
y
y
y
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
.
5
+
y
y
y
y
0
.
5
+
0
.
5
-
y
y
x
x
0
.
5
+
0.5
x
x
x
x
0
.
5
+
0
.
5
+
max
⁡
(
x
,
y
)
max
⁡
(
x
,
y
)
max
⁡
(
x
,
y
)
1Theorem 15.
Letμ be the normalized capacity on N={1,2} such that μ1=0 and μ2=0.5. If U∈{U⊥,U⊤,Umin,Umax}, then there do not exist weighting vectors p and w such that μ=υp,wU.Proof.
GivenU∈{U⊥,U⊤,Umin,Umax}, consider two weighting vectors p and w such that U(p2,w1)=0.5. By Lemma 14, we have p2=w1=0.5. Therefore, U(p1,w1)=U(0.5,0.5)=0.5 and, consequently, U(p1,w1)=0 is not possible.In each of the following theorems we consider the semiuninormsUTL, UTM, UP~, and UP, respectively, and we show that any normalized capacity can be written as a SUOWA operator associated with appropriate weighting vectors p and w, which are given explicitly.Theorem 16.
Letμ be a normalized capacity on N={1,2} and let p and w be two weighting vectors defined as follows:(1)
Ifμ1+μ2<1, then (16)p=0.5+μ1-μ22,0.5+μ2-μ12,w=μ1+μ22,1-μ1+μ22.(2)
Ifμ1+μ2≥1 and min(μ1,μ2)<0.5, then (17)p=μ1,1-μ1ifμ1>0.5,1-μ2,μ2ifμ2>0.5,w=μ1+μ2-0.5,1.5-μ1-μ2.(3)
Ifmin(μ1,μ2)≥0.5, then (18)p=0.5+μ1-μ2,0.5+μ2-μ1,w=max⁡μ1,μ2,1-max⁡μ1,μ2.Then, μ=υp,wUTL; that is, Cμ=Sp,wUTL.Proof.
Letμ be a normalized capacity on N={1,2} and recall that when n=2, the semiuninorm UTL is defined by(19)UTLx,y=max⁡x,yifx,y∈0.5,12,max⁡x+y-0.5,0otherwise.We distinguish the following cases:
(1) Ifμ1+μ2<1, consider (20)p=0.5+μ1-μ22,0.5+μ2-μ12,w=μ1+μ22,1-μ1+μ22.Then,(21)UTLp1,w1=0.5+μ1-μ22+μ1+μ22-0.5=μ1,UTLp2,w1=0.5+μ2-μ12+μ1+μ22-0.5=μ2.
(2) Ifμ1+μ2≥1 and min(μ1,μ2)<0.5, consider (22)p=μ1,1-μ1ifμ1>0.5,1-μ2,μ2ifμ2>0.5,w=μ1+μ2-0.5,1.5-μ1-μ2.We distinguish two cases:(a)
Ifμ1>0.5, then (23)UTLp1,w1=max⁡μ1,μ1+μ2-0.5=μ1,UTLp2,w1=1-μ1+μ1+μ2-0.5-0.5=μ2.(b)
Ifμ2>0.5, then (24)UTLp1,w1=1-μ2+μ1+μ2-0.5-0.5=μ1,UTLp2,w1=max⁡μ2,μ1+μ2-0.5=μ2.
(3) Ifmin(μ1,μ2)≥0.5, consider (25)p=0.5+μ1-μ2,0.5+μ2-μ1,w=max⁡μ1,μ2,1-max⁡μ1,μ2.We distinguish three cases:(a)
Ifμ1>μ2, then (26)UTLp1,w1=max⁡0.5+μ1-μ2,μ1=μ1,UTLp2,w1=0.5+μ2-μ1+μ1-0.5=μ2.(b)
Ifμ1=μ2, then (27)UTLp1,w1=max⁡0.5,μ1=μ1,UTLp2,w1=max⁡0.5,μ2=μ2.(c)
Ifμ1<μ2, then (28)UTLp1,w1=0.5+μ1-μ2+μ2-0.5=μ1,UTLp2,w1=max⁡0.5+μ2-μ1,μ2=μ2.Theorem 17.
Letμ be a normalized capacity on N={1,2} and let p and w be two weighting vectors defined as follows:(1)
Ifmax(μ1,μ2)<0.5, then (29)p=0.5+μ1-μ2,0.5+μ2-μ1,w=min⁡μ1,μ2,1-min⁡μ1,μ2.(2)
Ifμ1+μ2<1 and max(μ1,μ2)≥0.5, then (30)p=μ1,1-μ1ifμ1<0.5,1-μ2,μ2ifμ2<0.5,w=μ1+μ2-0.5,1.5-μ1-μ2.(3)
Ifμ1+μ2≥1 and min(μ1,μ2)<0.5, then (31)p=μ1,1-μ1ifμ1>0.5,1-μ2,μ2ifμ2>0.5,w=μ1+μ2-0.5,1.5-μ1-μ2.(4)
Ifmin(μ1,μ2)≥0.5, then (32)p=0.5+μ1-μ2,0.5+μ2-μ1,w=max⁡μ1,μ2,1-max⁡μ1,μ2.Then, μ=υp,wUTM; that is, Cμ=Sp,wUTM.Proof.
Letμ be a normalized capacity on N={1,2} and recall that when n=2, the semiuninorm UTM is defined by(33)UTMx,y=max⁡x,yifx,y∈0.5,12,min⁡x,yifx,y∈0,0.52,x+y-0.5otherwise.We distinguish the following cases:
(1) Ifmax(μ1,μ2)<0.5, consider (34)p=0.5+μ1-μ2,0.5+μ2-μ1,w=min⁡μ1,μ2,1-min⁡μ1,μ2.We distinguish three cases:(a)
Ifμ1<μ2, then (35)UTMp1,w1=min⁡0.5+μ1-μ2,μ1=μ1,UTMp2,w1=0.5+μ2-μ1+μ1-0.5=μ2.(b)
Ifμ1=μ2, then (36)UTMp1,w1=min⁡0.5,μ1=μ1,UTMp2,w1=min⁡0.5,μ2=μ2.(c)
Ifμ1>μ2, then (37)UTMp1,w1=0.5+μ1-μ2+μ2-0.5=μ1,UTMp2,w1=min⁡0.5+μ2-μ1,μ2=μ2.
(2) Ifμ1+μ2<1 and max(μ1,μ2)≥0.5, consider (38)p=μ1,1-μ1ifμ1<0.5,1-μ2,μ2ifμ2<0.5,w=μ1+μ2-0.5,1.5-μ1-μ2.
We distinguish two cases:(a)
Ifμ1<0.5, then (39)UTMp1,w1=min⁡μ1,μ1+μ2-0.5=μ1,UTMp2,w1=1-μ1+μ1+μ2-0.5-0.5=μ2.(b)
Ifμ2<0.5, then (40)UTMp1,w1=1-μ2+μ1+μ2-0.5-0.5=μ1,UTMp2,w1=min⁡μ2,μ1+μ2-0.5=μ2.
(3) Ifμ1+μ2≥1 and min(μ1,μ2)<0.5, then the proof of this case is similar to that of the second item in Theorem 16.
(4) Ifmin(μ1,μ2)≥0.5, then the proof of this case is similar to that of the third item in Theorem 16.Theorem 18.
Letμ be a normalized capacity on N={1,2} and let p and w be two weighting vectors defined as follows: (1)
Ifμ1+μ2<1, then (41)p=p1,p2∈W2ifμ1=μ2=0,μ1μ1+μ2,μ2μ1+μ2otherwise,w=μ1+μ22,1-μ1+μ22.(2)
Ifμ1+μ2≥1 and min(μ1,μ2)<2max(μ1,μ2)(1-max(μ1,μ2)), then (42)p=μ1,1-μ1ifμ1≥μ2,1-μ2,μ2ifμ1<μ2,w=min⁡μ1,μ221-max⁡μ1,μ2,1-min⁡μ1,μ221-max⁡μ1,μ2.(3)
Ifμ1+μ2≥1 and min(μ1,μ2)≥2max(μ1,μ2)(1-max(μ1,μ2)), then (43)p=1-μ22μ1,μ22μ1ifμ1≥μ2,μ12μ2,1-μ12μ2ifμ1<μ2,w=max⁡μ1,μ2,1-max⁡μ1,μ2.Then, μ=υp,wUP~; that is, Cμ=Sp,wUP~.Proof.
Letμ be a normalized capacity on N={1,2} and recall that when n=2, the semiuninorm UP~ is defined by (44)UP~x,y=max⁡x,yifx,y∈0.5,12,2xyotherwise.We distinguish the following cases:
(1) Ifμ1+μ2<1, consider (45)p=p1,p2∈W2ifμ1=μ2=0,μ1μ1+μ2,μ2μ1+μ2otherwise,w=μ1+μ22,1-μ1+μ22.We distinguish two cases:(a)
Ifμ1=μ2=0, then (46)UP~p1,w1=2·p1·0=0=μ1,UP~p2,w1=2·p2·0=0=μ2.(b)
If(μ1,μ2)≠(0,0), then (47)UP~p1,w1=2μ1μ1+μ2μ1+μ22=μ1,UP~p2,w1=2μ2μ1+μ2μ1+μ22=μ2.
(2) Ifμ1+μ2≥1 and min(μ1,μ2)<2max(μ1,μ2)(1-max(μ1,μ2)), then notice that the case max(μ1,μ2)=1 is not possible. Moreover, we have (48)min⁡μ1,μ221-max⁡μ1,μ2<max⁡μ1,μ2.On the other hand, given that min(μ1,μ2)≥1-max(μ1,μ2), we get(49)min⁡μ1,μ221-max⁡μ1,μ2≥0.5,and, consequently, max(μ1,μ2)>0.5. Now consider the following weighting vectors: (50)p=μ1,1-μ1ifμ1≥μ2,1-μ2,μ2ifμ1<μ2,w=min⁡μ1,μ221-max⁡μ1,μ2,1-min⁡μ1,μ221-max⁡μ1,μ2.We distinguish two cases:(a)
Ifμ1≥μ2, then (51)UP~p1,w1=max⁡μ1,μ221-μ1=μ1,UP~p2,w1=21-μ1μ221-μ1=μ2.(b)
Ifμ1<μ2, then (52)UP~p1,w1=21-μ2μ121-μ2=μ1,UP~p2,w1=max⁡μ2,μ121-μ2=μ2.
(3) Ifμ1+μ2≥1 and min(μ1,μ2)≥2max(μ1,μ2)(1-max(μ1,μ2)), then max(μ1,μ2)≥0.5, and we also have(53)min⁡μ1,μ22max⁡μ1,μ2≥1-max⁡μ1,μ2,or, equivalently,(54)1-min⁡μ1,μ22max⁡μ1,μ2≤max⁡μ1,μ2.On the other hand, since min(μ1,μ2)≤max(μ1,μ2), we get(55)min⁡μ1,μ22max⁡μ1,μ2≤0.5,and, consequently,(56)1-min⁡μ1,μ22max⁡μ1,μ2≥0.5.Consider now the following weighting vectors: (57)p=1-μ22μ1,μ22μ1ifμ1≥μ2,μ12μ2,1-μ12μ2ifμ1<μ2,w=max⁡μ1,μ2,1-max⁡μ1,μ2.We distinguish three cases:(a)
Ifμ1>μ2, then (58)UP~p1,w1=max⁡1-μ22μ1,μ1=μ1,UP~p2,w1=2μ22μ1μ1=μ2.(b)
Ifμ1=μ2, then (59)UP~p1,w1=max⁡0.5,μ1=μ1,UP~p2,w1=max⁡0.5,μ2=μ2.(c)
Ifμ1<μ2, then (60)UP~p1,w1=2μ12μ2μ2=μ1,UP~p2,w1=max⁡1-μ12μ2,μ2=μ2.Theorem 19.
Letμ be a normalized capacity on N={1,2} and let p and w be two weighting vectors defined as follows: (1)
Ifμ1+μ2<1 and max(μ1,μ2)<2min(μ1,μ2)(1-min(μ1,μ2)), then (61)p=1-μ22μ1,μ22μ1ifμ1≤μ2,μ12μ2,1-μ12μ2ifμ1>μ2,w=min⁡μ1,μ2,1-min⁡μ1,μ2.(2)
Ifμ1+μ2<1 and max(μ1,μ2)≥2min(μ1,μ2)(1-min(μ1,μ2)), then (62)p=μ1,1-μ1ifμ1≤μ2,1-μ2,μ2ifμ1>μ2,w=max⁡μ1,μ221-min⁡μ1,μ2,1-max⁡μ1,μ221-min⁡μ1,μ2.(3)
Ifμ1+μ2≥1 and min(μ1,μ2)<2max(μ1,μ2)(1-max(μ1,μ2)), then (63)p=μ1,1-μ1ifμ1≥μ2,1-μ2,μ2ifμ1<μ2,w=min⁡μ1,μ221-max⁡μ1,μ2,1-min⁡μ1,μ221-max⁡μ1,μ2.(4)
Ifμ1+μ2≥1 and min(μ1,μ2)≥2max(μ1,μ2)(1-max(μ1,μ2)), then (64)p=1-μ22μ1,μ22μ1ifμ1≥μ2,μ12μ2,1-μ12μ2ifμ1<μ2,w=max⁡μ1,μ2,1-max⁡μ1,μ2.Then, μ=υp,wUP; that is, Cμ=Sp,wUP.Proof.
Letμ be a normalized capacity on N={1,2} and recall that when n=2, the semiuninorm UP is defined by (65)UPx,y=max⁡x,yifx,y∈0.5,12,min⁡x,yifx,y∈0,0.52,2xyotherwise.We distinguish the following cases:
(1) Ifμ1+μ2<1 and max(μ1,μ2)<2min(μ1,μ2)(1-min(μ1,μ2)), then notice that the case min(μ1,μ2)=0 is not possible. Moreover, min(μ1,μ2)<0.5, and we also have(66)max⁡μ1,μ22min⁡μ1,μ2<1-min⁡μ1,μ2,or, equivalently,(67)min⁡μ1,μ2<1-max⁡μ1,μ22min⁡μ1,μ2.On the other hand, since min(μ1,μ2)≤max(μ1,μ2), we get(68)max⁡μ1,μ22min⁡μ1,μ2≥0.5,and, consequently,(69)1-max⁡μ1,μ22min⁡μ1,μ2≤0.5.Consider now the following weighting vectors: (70)p=1-μ22μ1,μ22μ1ifμ1≤μ2,μ12μ2,1-μ12μ2ifμ1>μ2,w=min⁡μ1,μ2,1-min⁡μ1,μ2.We distinguish three cases:(a)
Ifμ1<μ2, then (71)UPp1,w1=min⁡1-μ22μ1,μ1=μ1,UPp2,w1=2μ22μ1μ1=μ2.(b)
Ifμ1=μ2, then (72)UPp1,w1=min⁡0.5,μ1=μ1,UPp2,w1=min⁡0.5,μ2=μ2.(c)
Ifμ1>μ2, then (73)UPp1,w1=2μ12μ2μ2=μ1,UPp2,w1=min⁡1-μ12μ2,μ2=μ2.
(2) Ifμ1+μ2<1 and max(μ1,μ2)≥2min(μ1,μ2)(1-min(μ1,μ2)), then notice that the case min(μ1,μ2)=1 is not possible. Moreover, we have (74)max⁡μ1,μ221-min⁡μ1,μ2≥min⁡μ1,μ2.On the other hand, given that max(μ1,μ2)<1-min(μ1,μ2), we get(75)max⁡μ1,μ221-min⁡μ1,μ2<0.5,and, consequently, min(μ1,μ2)<0.5. Now consider the following weighting vectors: (76)p=μ1,1-μ1ifμ1≤μ2,1-μ2,μ2ifμ1>μ2,w=max⁡μ1,μ221-min⁡μ1,μ2,1-max⁡μ1,μ221-min⁡μ1,μ2.We distinguish two cases: (a)
Ifμ1≤μ2, then (77)UPp1,w1=min⁡μ1,μ221-μ1=μ1,UPp2,w1=21-μ1μ221-μ1=μ2.(b)
Ifμ1>μ2, then (78)UPp1,w1=21-μ2μ121-μ2=μ1,UPp2,w1=minμ2,μ121-μ2=μ2.
(3) Ifμ1+μ2≥1 and min(μ1,μ2)<2max(μ1,μ2)(1-max(μ1,μ2)), then the proof of this case is similar to that of the second item in Theorem 18.
(4) Ifμ1+μ2≥1 and min(μ1,μ2)≥2max(μ1,μ2)(1-max(μ1,μ2)), then the proof of this case is similar to that of the third item in Theorem 18.


## 5. Conclusion

SUOWA operators are a useful tool for dealing with situations where combining values by using both a weighted mean and an OWA type aggregation is necessary. Given that they are Choquet integral-based operators with respect to normalized capacities, they have some natural properties such as continuity, monotonicity, idempotency, compensativeness, and homogeneity of degree 1. For this reason, it seems interesting to analyze their behavior from different points of view. In this paper, we have shown that, in two dimensions, if we consider one of the following continuous semiuninorms:UTL, UTM, UP~, and UP, then any Choquet integral with respect to a normalized capacity can be expressed as a SUOWA operator associated with the chosen semiuninorm and two weighting vectors p and w, which are given explicitly.



---

*Source: 271491-2015-08-02.xml*