CVAR REDUCED FUZZY VARIABLES AND THEIR SECOND ORDER MOMENTS

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1 Iranian Journal of Fuzzy Systems Vol., No. 5, (05 pp CVAR REDUCED FUZZY VARIABLES AND THEIR SECOND ORDER MOMENTS X. J. BAI AND Y. K. LIU Abstract. Based on credibilistic value-at-risk (CVaR of regular fuzzy variable, we introduce a new CVaR reduction method for type- fuzzy variables. The reduced fuzzy variables are characterized by parametric possibility distributions. We establish some useful analytical epressions for mean values and second order moments of common reduced fuzzy variables. The conve properties of second order moments with respect to parameters are also discussed. Finally, we take second order moment as a new risk measure, and develop a mean-moment model to optimize fuzzy portfolio selection problems. According to the analytical formulas of second order moments, the mean-moment optimization model is equivalent to parametric quadratic conve programming problems, which can be solved by general-purpose optimization software. The solution results reported in the numerical eperiments demonstrate the credibility of the proposed optimization method.. Introduction As a natural etension of type- fuzzy set, type- fuzzy set was first introduced by Zadeh [9. Since then, type- fuzzy set theory has been further eplored in the literature. Dubois and Prade [3 investigated the operations in a fuzzy-valued logic. Mizumoto and Tanaka [4 discussed what kinds of algebraic structures the grades of type- fuzzy sets form under join, meet and negation. Mendel [ summarized the developments and applications of type- fuzzy sets before the year 00. Mendel and John [3 introduced some basic concepts to characterize type- fuzzy set, including the type- membership function, the secondary membership function, the footprint of uncertainty and so on. Hu and Wang [5 introduced interval-valued type- fuzzy sets and interval-valued type- fuzzy relations and discussed their properties. In the literature, type- fuzzy numbers have been used to indicate the similarity degree of fuzzy sets [, 7. At the same time, type- fuzzy sets have been applied successfully to many application-oriented fields. For eample, Kundu et al. [ considered two fied charge transportation problems with type- fuzzy parameters. Huang et al. [6 proposed a novel dynamic optimal path planning method, which employed a type- fuzzy logic inference system to path analysis for each cell established in the cellular automata algorithm. For recent developments about type- fuzzy theory, Received: June 03; Revised: July 04; Accepted: June 05 Key words and phrases: Credibilistic value-at-risk, Reduced fuzzy variable, Parametric possibility distribution, Second order moment.

2 46 X. J. Bai and Y. K. Liu the interested readers may refer to Hao and Mendel [4, Ngan [6, Moharrer et al. [5 and Torshizi and Zarandi [7. Defuzzification is one of the critical steps involved in type- fuzzy system. By reduction, all computations in three-dimensional space are degenerated into calculations in two-dimensional plane so as to reduce greatly the computing compleity. Several reduction methods for type- fuzzy sets have been presented in the literature. Coupland and John [ presented a geometric-based defuzzication method for type- fuzzy sets. Liu [0 gave a centroid type-reduction strategy for interval type- fuzzy sets. Motivated by the work mentioned above, the present paper introduces a new reduction method in fuzzy possibility theory. First, we define the CVaR for regular fuzzy variables based on credibility measure. Then, we develop CVaR reduction method for secondary possibility distributions. The idea of the proposed method is to reduce uncertainty in secondary possibility distributions, and retains the most important information in the parametric possibility distributions of reduced fuzzy variables. There are two types of parameters included in the obtained parametric possibility distributions. The first type of parameters is to describe the degree of uncertainty that a type- fuzzy variable takes on its values, while the second type of parameters is to represent the credibility level in the support of type- fuzzy variable. From the geometrical viewpoint, the first type of parameters determines the shape of the support of a type- fuzzy variable, while the second type of parameters determines the location of possibility distribution in the support of the type- fuzzy variable. From this viewpoint, our CVaR reduction method has some advantages over other eisting methods by introducing the location parameter in the possibility distribution. Since the CVaR reduced fuzzy variables have parametric possibility distributions, the computation about their numerical characteristics is an interesting issue for research. In this paper, we first derive the analytical epressions of mean values for common reduced fuzzy variables. Then we derive the analytical epressions of the second order moments to measure the variations of parametric possibility distributions with respect to mean values. Finally, we we take mean value and second order moment as two optimization indees, and apply them to fuzzy portfolio selection problems. The rest of this paper is organized as follows. Section introduces some concepts in fuzzy theory. Section 3 defines the CVaR for regular fuzzy variable, and derives some useful CVaR formulas for common regular fuzzy variables. In Section 4, we develop the CVaR reduction method for type- fuzzy variables. For common reduced fuzzy variables, Section 5 discusses the computation of mean values, and Section 6 establishes the analytical epressions of the second order moments. Section 7 provides a practical application eample about portfolio selection problem, in which the proposed second order moment is taken as a new measure to gauge the risk resulted from fuzzy uncertainty. Finally, Section concludes the paper.. Fundamental Concepts Let (Γ, A, Pos be a possibility space, where Γ is the universe of discourse, A an ample field on Γ that is closed under arbitrary unions, intersections and complement, and Pos a possibility measure on A. Using possibility measure, the credibility

3 CVaR Reduced Fuzzy Variables and Their Second Order Moments 47 measure of an event A A was defined as Cr(A = ( + Pos(A Pos(Ac, where A c = Γ\A is the complementary event of A. Definition.. [ Let (Γ, A be an ample space. A function ξ : Γ R is called a fuzzy variable if γ ξ(γ r} A for any r R. Let (Γ, A, Pos be a possibility space. An m-ary regular fuzzy vector ξ = (ξ,..., ξ m is defined as a measurable map from Γ to the space [0, m in the sense that for every r = (r,..., r m [0, m, one has γ Γ ξ(γ r} = γ Γ ξ (γ r, ξ (γ r,..., ξ m (γ r m } A. When m =, ξ is called a regular fuzzy variable (RFV. In this paper, we denote R([0, as the collection of all RFVs on [0,. If ξ = (r, r, r 3 with 0 r < r < r 3, then ξ is a triangular RFV. Similarly, if ξ = (r, r, r 3, r 4 with 0 r < r r 3 < r 4, then ξ is a trapezoidal RFV. Definition.. [9 Let ξ be a fuzzy variable defined on a possibility space (Γ, A, Pos. The credibility distribution function of ξ is defined by G ξ (r = Crγ Γ ξ(γ r}, r R. Eample.3. Let ξ = (0., 0.4, 0.5, 0.5 be a trapezoidal RFV. The possibility distribution of ξ is shown in Figure. The credibility distribution of ξ is computed by 0, if < 0. 5r, if 0. r 0.4 Crξ r} = which is plotted in Figure. 0r 3, if 0.4 r 0.5 4, if , if > 0.5, Let Pos : A R([0, be a set function defined on A such that Pos(A A A atom} is a family of mutually independent RFVs. We call Pos a fuzzy possibility measure if it satisfies the following conditions: ( P : Pos( = 0; ( P : For any subclass A i i I} of A (finite, countable or uncountable ( Pos A i = sup Pos(A i. i I i I Moreover, if µ Pos(Γ ( =, then we call Pos a normalized fuzzy possibility measure. The triplet (Γ, A, Pos is referred to as a fuzzy possibility space (FPS.

4 4 X. J. Bai and Y. K. Liu Figure. The Possibility Distribution of ξ Figure. The Credibility Distribution of ξ Remark.4. Fuzzy possibility measure is a generalization of (non-fuzzy possibility measure in the literature. That is, if for any A A, Pos(A is a crisp number in [0, instead of a fuzzy number in [0,, then Pos is just a (non-fuzzy possibility measure, and denoted by Pos. For the sake of clarity, we provide the following eample to eplain the difference between possibility measure and fuzzy possibility measure. Let Γ = γ, γ, γ 3 } and A = P(Γ, the power set of Γ. Define a set function Pos on P(Γ as follows: Posγ } = 0., Posγ } =, Posγ 3 } = 0.6, and for any other set A P(Γ, Pos(A = ma γi A Posγ i }. Then Pos is a possibility measure, and (Γ, A, Pos is a possibility space.

5 CVaR Reduced Fuzzy Variables and Their Second Order Moments 49 On the other hand, if we define a set function Pos : P(Γ R([0, as Posγ } = (0.5, 0., 0.4, Posγ } =, Posγ 3 } = (0.5, 0.6, 0.7, and for any other set A P(Γ, Pos(A = ma γi A Posγ i }, then Pos is a fuzzy possibility measure, and (Γ, A, Pos is a fuzzy possibility space. Let (Γ, A, Pos be an FPS. A map ξ = ( ξ, ξ,..., ξ m : Γ R m is called an m-ary type- fuzzy vector if for any r = (r, r,..., r m R m, the set γ Γ ξ(γ r} is an element of A, i.e., γ Γ ξ(γ r} = γ Γ ξ (γ r, ξ (γ r,..., ξ m (γ r m } A. As m =, the map ξ : Γ R is called a type- fuzzy variable. Let ξ = ( ξ, ξ,..., ξ m be a type- fuzzy vector. The secondary possibility distribution function µ ξ( of ξ is a map R m R([0, such that µ ξ( = Posγ Γ ξ(γ = }, R m, while the type- possibility distribution function µ ξ( of ξ is a map R m J [0, such that µ ξ(, u = Pos µ ξ( = u}, (, u R m J, where Pos is the possibility measure induced by the distribution of µ ξ(, and J [0, is the support of µ ξ(, i.e., J = u [0, µ ξ(, u > 0}. The support of a type- fuzzy vector ξ is defined as supp ξ = (, u R m [0, µ ξ(, u > 0}, where µ ξ(, u is the type- possibility distribution function of ξ. Eample.5. Let ξ = ( 3, 5, 9; 0.5, 0. be a type- triangular fuzzy variable. The support of ξ is shown in Figure 3. Figure 3. The Support of Type- Triangular Fuzzy Variable ξ

6 50 X. J. Bai and Y. K. Liu 3. The CVaRs for Regular Fuzzy Variables If ξ is a regular fuzzy variable, then the CVaR of ξ, denoted by CVaR α (ξ, is defined by CVaR α (ξ = infr Crξ r} α}, α (0,. The CVaR of ξ is different from the lower VaR and upper VaR defined by possibility measure for a regular fuzzy variable. When a regular fuzzy variable ξ has continuous possibility distribution, we have the following relations among the CVaR, lower VaR and upper VaR. For any α (0,, we have CVaR α (ξ = VaRL α(ξ, and CVaR α (ξ = VaRU α (ξ. In the following, we derive some useful VaR formulas for common regular fuzzy variables. Theorem 3.. If ξ is a triangular regular fuzzy variable, then we have r + α(r CVaR α (ξ = r, if α (0, 0.5 r r 3 + α(r 3 r, if α (0.5,. Proof. According to the possibility distribution of ξ, we have the following credibility distribution of ξ, 0, if < r r (r Crξ } =, if r r r r 3 r + (r, if r 3 r r 3, if > r 3. If α (0, 0.5, then CVaR α (ξ is the solution of the following equation r (r r α = 0. Therefore, we have CVaR α (ξ = r + α(r r. On the other hand, if α (0.5,, then CVaR α (ξ is the solution of the following equation r 3 r + α = 0. (r 3 r Thus, we have CVaR α (ξ = r r 3 + α(r 3 r. The proof of theorem is complete. As a consequence of Theorem 3., we have the following results about the relations among the VaRs, upper mean value E, lower mean vale E and mean value E of regular triangular fuzzy variable. Corollary 3.. Let ξ = (r 0 θ l, r 0, r 0 + θ r be a triangular regular fuzzy variable with θ l, θ r > 0. (i If α = 3/4, then CVaR α (ξ = E [ξ; (ii If α = /4, then CVaR α (ξ = E [ξ; (iii If θ l θ r and α = (5θ r θ l /θ r, then CVaR α (ξ = E[ξ; (iv If θ l θ r and α = (3θ l + θ r /θ l, then CVaR α (ξ = E[ξ.

7 CVaR Reduced Fuzzy Variables and Their Second Order Moments 5 Theorem 3.3. If ξ is a trapezoidal regular fuzzy variable, then we have r + α(r CVaR α (ξ = r, if α (0, 0.5 r 3 r 4 + α(r 4 r 3, if α (0.5,. Proof. First, the credibility distribution of fuzzy variable ξ is calculated by 0, if < r r (r, if r r r Crξ } =, if r r 3 r 4 r 3+ (r, if r 4 r 3 3 r 4, if > r 4. If α (0, 0.5, then CVaR α (ξ is the solution of the following equation r (r r α = 0. Thus, we have CVaR α (ξ = r + α(r r. On the other hand, if α (0.5,, then CVaR α (ξ is the solution of the following equation r 4 r 3 + α = 0. (r 4 r 3 Thus, we have CVaR α (ξ = r 3 r 4 + α(r 4 r 3. The proof of theorem is complete. Theorem 3.4. Let ξ be a normal regular fuzzy variable with the following possibility distribution ( ( µ µ ξ ( = ep σ, [0,, µ [0,. If we denote a = ep( µ /σ and b = ep( ( µ /σ, then we have µ σ ln α, if α [ a CVaR α (ξ =, 0.5 µ + σ ln ( α, if α ( 0.5, b. Proof. The credibility distribution of fuzzy variable ξ is the following function ( Crξ } = ep, ( µ if 0 µ ( σ ep ( µ σ, if µ. If α (a/, 0.5, then CVaR α (ξ is the solution of the following equation ( ep ( µ σ α = 0, 0 µ. Hence, we have CVaR α (ξ = µ σ ln α. On the other hand, if α (0.5, b/, then CVaR α (ξ is the solution of the following equation ( ep ( µ σ α = 0, µ. Thus, we have CVaR α (ξ = µ + σ ln ( α. The proof of theorem is complete.

8 5 X. J. Bai and Y. K. Liu Theorem 3.5. Let ξ be a gamma regular fuzzy variable with the following possibility distribution ( r ( µ ξ ( = ep r, [0,, λr λ where r is a positive integer, 0 < λ /r, and denote c = (/λr r ep(r /λ. (i If α (0, 0.5, then CVaR α (ξ of ξ is the solution of the following equation r ( ep r ( α = 0, [0, λr; λr λ (ii If α (0.5, c/, then CVaR α (ξ of ξ is the solution of the following equation ( r ( ep r α = 0, [λr,. λr λ Proof. We only prove assertion (i, and assertion (ii can be proved similarly. According to the possibility distribution of ξ, we have ( r ( λr ep r λ, if 0 λr Crξ } = ( r ( λr ep r λ, if λr. By the definition of CVaR α (ξ, we know that CVaR α (ξ is the solution of the following equation r ( ep r ( α = 0, [0, λr, λr λ which completes the proof of assertion (i. 4. A New CVaR Reduction Method Let (Γ, A, Pos be a fuzzy possibility space, and ξ a type- fuzzy variable with secondary possibility distribution µ ξ(. To reduce the uncertainty in µ ξ(, we employ the CVaR of Pos ξ = } as the representing value of µ ξ(. The method is referred to as the CVaR reduction. The reduced fuzzy variable obtained by CVaR reduction method is denoted by ξ. There are two kinds of parameters included in the possibility distributions of CVaR reduced fuzzy variables. The first type parameters θ l and θ r represent the degree of uncertainty that a type- fuzzy variable ξ takes on its value, while the second type parameter α means the credibility level in the support of a type- fuzzy variable. From the geometrical viewpoint, the parameters θ l and θ r determine the lower and upper boundaries of possibility distribution, while α determines the location of possibility distribution between the lower boundary and upper boundary. When parameter α varies in the interval (0,, the possibility distribution varies between the lower and upper boundaries. As a consequence, our CVaR reduction method is more fleible than other eisting reduction methods by introducing the parameter α in possibility distributions. In the following, we discuss the CVaR reduction for common type- fuzzy variables.

9 CVaR Reduced Fuzzy Variables and Their Second Order Moments 53 Theorem 4.. Let ξ = ( r, r, r 3 ; θ l, θ r be a type- triangular fuzzy variable, and θ = (θ l, θ r. (i If α (0, 0.5, then the reduced fuzzy variable ξ has the following parametric possibility distribution ( θ l + αθ l r r r, if [r, r+r (+θ l αθ l ( αθ l r r r µ ξ (; θ, α = r, if [ r+r, r (+θ l αθ l +( αθ l r +r 3 r 3 r, if [r, r+r3 ( θ l + αθ l r3 r 3 r, if [ r+r3, r 3. (ii If α (0.5,, then the reduced fuzzy variable ξ has the following parametric possibility distribution ( θ r + αθ r r r r, if [r, r+r (+θ r αθ r ( αθ rr r r µ ξ (; θ, α = r, if [ r+r, r (+θ r αθ r+( αθ rr +r 3 r 3 r, if [r, r+r3 ( θ r + αθ r r3 r 3 r, if [ r+r3, r 3. Proof. We only prove the second assertion, and the first can be proved similarly. Note that the secondary possibility distribution µ ξ( of ξ is the following triangular regular fuzzy variable ( r r r θ l min for any [r, r, and ( r 3 θ l min r 3 r r r r, r r r r3 r 3 r, r r 3 r }, r, r r + θ r min, r } r r r r r r r r }, r3, r3 r3 + θ r min, } r r 3 r r 3 r r 3 r r 3 r for any [r, r 3. Since ξ is the CVaR reduced fuzzy variable of ξ, we have µ ξ (; θ, α = Posξ = } } r r r = r ( αθ l min r r, r r r, if [r, r } r 3 r r 3 r ( αθ l min 3 r 3 r, r r 3 r, if [r, r 3 ( θ l + αθ l r r r, if [r, r+r (+θ l αθ l ( αθ l r r = r r, if [ r+r, r (+θ l αθ l +( αθ l r +r 3 r 3 r, if [r, r+r3 ( θ l + αθ l r3 r 3 r, if [ r+r3, r 3, which completes the proof of assertion (ii. The following corollary shows that the E, E and E reduction methods are the special cases of the CVaR reduction method for type- triangular fuzzy variable. Corollary 4.. Let ξ be a type- triangular fuzzy variable and ξ, ξ and ξ 3 be the reduced fuzzy variables obtained by E, E and E reduction methods respectively. (i For E reduction method, µ ξ (; θ, 3 4 = µ ξ (; θ;

10 54 X. J. Bai and Y. K. Liu (ii For E reduction method, µ ξ (; θ, 4 = µ ξ (; θ; (iii For E reduction method, if θ l θ r, then µ ξ (; θ, 5θr θ l θ r = µ ξ3 (; θ; (iv For E reduction method, if θ l θ r, then µ ξ (; θ, 3θ l+θ r θ l = µ ξ3 (; θ. Eample 4.3. Let ξ = (, 3, 4; 0.5, be a type- triangular fuzzy variable. Suppose ξ is the CVaR reduced fuzzy variable of ξ. By the CVaR reduction method, if α (0, 0.5, the reduced fuzzy variable ξ of ξ has the following possibility distribution (0.5 + α α, if [, 5 (.5 α + 3α 3.5, if [ 5 µ ξ (; θ, α =, 3 (.5 α 3α + 5.5, if [3, 7 (0.5 + α + 4α +, if [ 7, 4, and if α (0.5,, the reduced fuzzy variable ξ of ξ has the following possibility distribution α 4α, if [, 5 ( α + 6α 5, if [ 5 µ ξ (; θ, α =, 3 ( α 6α + 7, if [3, 7 α + α, if [ 7, 4. Theorem 4.4. Let ξ = ( r, r, r 3, r 4 ; θ l, θ r be a type- trapezoidal fuzzy variable, and θ = (θ l, θ r. (i If α (0, 0.5, then the reduced fuzzy variable ξ has the following parametric possibility distribution ( θ l + αθ l r r r, if [r, r+r (+θ l αθ l ( αθ l r r r r, if [ r+r, r µ ξ (; θ, α =, if [r, r 3 (+θ l αθ l +( αθ l r 3+r 4 r 4 r 3, if [r 3, r3+r4 ( θ l + αθ l r4 r 4 r 3, if [ r3+r4, r 4. (ii If α (0.5,, then the reduced fuzzy variable ξ has the following parametric possibility distribution ( θ r + αθ r r r r, if [r, r+r (+θ r αθ r ( αθ rr r r r, if [ r+r, r µ ξ (; θ, α =, if [r, r 3 (+θ r αθ r+( αθ rr 3+r 4 r 4 r 3, if [r 3, r3+r4 ( θ r + αθ r r4 r 4 r 3, if [ r3+r4, r 4. Proof. We only prove assertion (i, and the rest can be proved similarly. Note that the secondary possibility distribution µ ξ( of ξ is the following triangular regular fuzzy variable ( r r θ l min, r }, r, r r + θ r min, r } r r r r r r r r r r r r r r

11 CVaR Reduced Fuzzy Variables and Their Second Order Moments 55 for any [r, r, the regular fuzzy variable for any [r, r 3, and ( r 4 r4 θ l min, } r3, r4, r4 r4 + θ r min, } r3 r 4 r 3 r 4 r 3 r 4 r 3 r 4 r 3 r 4 r 3 r 4 r 3 r 4 r 3 for any [r 3, r 4. Since ξ is the CVaR reduced fuzzy variable of ξ, we have µ ξ (; θ, α = Posξ = } r r r ( αθ l min = r 4 r 4 r 3 ( αθ l min ( θ l + αθ l r = } r r r, r r r, if [r, r, if [r, r 3, if [r 3, r 4 r r, r 4 r 4 r 3, r3 r 4 r 3 } if [r, r+r (+θ l αθ l ( αθ l r r r r, if [ r+r, r, if [r, r 3 (+θ l αθ l +( αθ l r 3+r 4 r 4 r 3, if [r 3, r3+r4 ( θ l + αθ l r4 r 4 r 3, if [ r3+r4, r 4, which completes the proof of assertion (i. Eample 4.5. Let ξ = (,, 3, 5; 0.6, 0. be a type- trapezoidal fuzzy variable. Suppose ξ is the CVaR reduced fuzzy variable of ξ. By the CVaR reduction method, if α (0, 0.5, the reduced fuzzy variable ξ of ξ has the following possibility distribution (0.4 +.α.α 0.4, if [, 3 (.6.α +.4α., if [ 3, µ ξ (; θ, α =, if [, 3 (0. 0.6α.α + 3.4, if [3, 4 ( α + 3α +, if [4, 5, and if α (0.5,, the reduced fuzzy variable ξ of ξ has the following possibility distribution (0. +.6α.6α 0., if [, 3 (..6α + 3.α.6, if [ 3, µ ξ (; θ, α =, if [, 3 (0.9 0.α.4α + 3.7, if [3, 4 ( α + 4α + 0.5, if [4, 5. Theorem 4.6. Let ξ = ñ(µ, σ ; θ l, θ r be a type- normal fuzzy variable, and θ = (θ l, θ r. (i If α (0, 0.5, then the reduced fuzzy variable ξ has the following parametric possibility distribution µ ξ (; θ, α = ( θ l + αθ l ep ( ( µ σ, if µ σ ln or µ + σ ln ( + θ l αθ l ep( ( µ σ ( αθ l, if µ σ ln µ + σ ln.

12 56 X. J. Bai and Y. K. Liu (ii If α (0.5,, then the reduced fuzzy variable ξ has the following parametric possibility distribution ( θ r + αθ r ep ( ( µ σ, if µ σ ln or µ + σ ln µ ξ (; θ, α = ( + θ r αθ r ep( ( µ σ ( αθ r, if µ σ ln µ + σ ln. Proof. We only prove assertion (i, and the rest can be proved similarly. Note that the secondary possibility distribution µ ξ( of ξ is the following triangular regular fuzzy variable ( ep ( ( ( µ ep ( ep ( µ σ θ l min σ, ( µ σ + θ r min ( ( ( µ ( } µ ep σ, ep σ, ( ( ( µ ( } µ ep σ, ep σ. Since ξ is the CVaR reduced fuzzy variable of ξ, we have µ ξ (; θ, α = Posξ = } ( ( µ = ep σ ( αθ l min ( θ l + αθ l ep ( ( µ σ, if µ σ ln or µ + σ ln = ( + θ l αθ l ep ( ( µ σ ( αθl, if µ σ ln µ + σ ln, which completes the proof of assertion (i. } ( µ ( µ ep( σ, ep( σ The following corollary shows that the E, E and E reduction methods are the special cases of the CVaR reduction method for type- normal fuzzy variable. Corollary 4.7. Let ξ be a type- normal fuzzy variable and η, η and η 3 be the reduced fuzzy variables obtained by E, E and E reduction methods respectively. (i For E reduction method, µ ξ (; θ, 3 4 = µ η (; θ; (ii For E reduction method, µ ξ (; θ, 4 = µ η (; θ; (iii For E reduction method, if θ l θ r, then µ ξ (; θ, 5θr θ l θ r = µ η3 (; θ; (iv For E reduction method, if θ l θ r, then µ ξ (; θ, 3θ l+θ r θ l = µ η3 (; θ. Eample 4.. Let ξ = ñ(, 0.5; 0.3, 0.7 be a type- normal fuzzy variable. Suppose ξ is the CVaR reduced fuzzy variable of ξ. By the CVaR reduction method, if α (0, 0.5, then the reduced fuzzy variable ξ of ξ has the following possibility distribution µ ξ (; θ, α =

13 CVaR Reduced Fuzzy Variables and Their Second Order Moments 57 ( α ep( (, if ln or + ln (.3 0.6α ep( ( 0.3( α, if ln + ln, and if α (0.5,, then the reduced fuzzy variable ξ of ξ has the following possibility distribution µ ξ (; θ, α = ( α ep( (, if ln or + ln (.7.4α ep( ( + 0.7(α, if ln + ln. Theorem 4.9. Let ξ = γ(λ, r; θ l, θ r be a type- gamma fuzzy variable, and θ = (θ l, θ r. (i If α (0, 0.5, then the reduced fuzzy variable ξ has the following parametric possibility distribution µ ξ (; θ, α = ( θl + αθ l ( λr ( + θ l αθ l ( λr r ( ( ep r λ, if r ( λr ep r λ r ( ep r λ ( αθl, if ( r ( λr ep r λ > (ii If α (0.5,, then the reduced fuzzy variable ξ has the following parametric possibility distribution µ ξ (; θ, α = ( θr + αθ r ( λr ( + θ r αθ r ( λr r ( ( ep r λ, if r ( λr ep r λ r ( ep r λ ( αθr, if ( r ( λr ep r λ > Proof. We only prove assertion (i, and the rest can be proved similarly. Note that the secondary possibility distribution µ ξ( of ξ is the following triangular regular fuzzy variable (( r ( ep r ( r ( θ l min ep r, ( λr λ λr λ λr r ep(r λ }, ( λr r ep(r λ, ( λr r ep(r λ + θ r min ( r ( ep r ( r (, ep r }. λr λ λr λ Since ξ is the CVaR reduced fuzzy variable of ξ, we have ( r ( µ ξ (; θ, α = Posξ = } = ep r ( λr λ r ( ( αθ l min ep r ( r (, ep r } λr λ λr λ ( θl + αθ = l ( r ( ( λr ep r λ, if r ( λr ep r λ ( + θ l αθ l ( r ( λr ep r λ ( αθl, if ( r ( λr ep r λ >, which completes the proof of assertion (i. The following corollary illustrates that the E, E and E reduction methods are the special cases of the CVaR reduction method for type- gamma fuzzy variable...

14 5 X. J. Bai and Y. K. Liu Corollary 4.0. Let ξ be a type- gamma fuzzy variable and ζ, ζ and ζ 3 be the reduced fuzzy variables obtained by E, E and E reduction methods respectively. (i For E reduction method, µ ξ (; θ, 3 4 = µ ζ (; θ; (ii For E reduction method, µ ξ (; θ, 4 = µ ζ (; θ; (iii For E reduction method, if θ l θ r, then µ ξ (; θ, 5θr θ l θ r = µ ζ3 (; θ; (iv For E reduction method, if θ l θ r, then µ ξ (; θ, 3θ l+θ r θ l = µ ζ3 (; θ. Eample 4.. Let ξ = γ(5, ; 0.5, 0. be a type- gamma fuzzy variable. Suppose ξ is the CVaR reduced fuzzy variable of ξ. By the CVaR reduction method, if α (0, 0.5, then the reduced fuzzy variable ξ of ξ has the following possibility distribution µ ξ (; θ, α = 00 ( + α ep ( ( 5, if ( 0 ep 5 00 (3 α ep ( 5 ( α, if ( ( 0 ep 5 >, and if α (0.5,, then the reduced fuzzy variable ξ of ξ has the following possibility distribution µ ξ (; θ, α = 500 ( + α ep ( ( 5, if ( 0 ep (9 α ep ( (α, if ( ( 0 ep 5 >. 5. The Mean Values of CVaR Reduced Fuzzy Variables For common CVaR reduced fuzzy variables, this section will derive the analytical epressions of mean values. Theorem 5.. Let ξ = ( r, r, r 3 ; θ l, θ r be a type- triangular fuzzy variable. Then the mean value of the CVaR reduced fuzzy variable ξ is E[ξ = r+r +r 3 4 r r+r3 r +r +r 3 4 r r+r3 ( αθ l, if α (0, 0.5 ( αθ r, if α (0.5,. Proof. We only prove the assertion in the case α (0, 0.5. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.. Thus E[ξ = 0 (ξ inf (β + ξ sup (β dβ. Note that µ ξ ((r + r / = µ ξ ((r + r 3 / = ( θ l + αθ l /. If β (0, ( θ l + αθ l /, then ξ inf (β and ξ sup (β are the solutions of the following equations respectively, ( θ l + αθ l r r r = β

15 CVaR Reduced Fuzzy Variables and Their Second Order Moments 59 and ( θ l + αθ l r 3 = β. r 3 r As a consequence, we have the following solutions and ξ inf (β = (r r β + r ( θ l + αθ l θ l + αθ l ξ sup (β = (r 3 r β + r 3 ( θ l + αθ l θ l + αθ l. Therefore, when β (0, ( θ l + αθ l /, we have ξ inf (β + ξ sup (β = (r r r 3 β + ( θ l + αθ l (r + r 3 θ l + αθ l. On the other hand, when β (( θ l + αθ l /,, it is similar to prove the following result ξ inf (β + ξ sup (β = (r r r 3 β + ( αθ l r + (r + r 3 + θ l αθ l. Hence, the mean value of ξ is computed by E[ξ = ( θ l +αθ l 0 (ξ inf (β + ξ sup (β dβ + = r + r + r 3 r r + r 3 ( αθ l. 4 The proof of theorem is complete. θ l +αθ l (ξ inf (β + ξ sup (β dβ Eample 5.. Let ξ be the type- triangular fuzzy variable defined in Eample 4.3. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 5., the mean value of ξ is E[ξ = 3. Theorem 5.3. Let ξ = ( r, r, r 3, r 4 ; θ l, θ r be a type- trapezoidal fuzzy variable. Then the mean value of the CVaR reduced fuzzy variable ξ is E[ξ = r+r +r 3+r 4 4 r r r3+r4 r +r +r 3+r 4 4 r r r3+r4 ( αθ l, if α (0, 0.5 ( αθ r, if α (0.5,. Proof. We only prove the assertion in the case α (0, 0.5. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.4. Note that µ ξ ((r + r / = µ ξ ((r 3 + r 4 / = ( θ l + αθ l /. If β (0, ( θ l + αθ l /, then ξ inf (β and ξ sup (β are the solutions of the following equations respectively, and ( θ l + αθ l r r r = β ( θ l + αθ l r 4 r 4 r 3 = β.

16 60 X. J. Bai and Y. K. Liu As a consequence, we have the following solutions and ξ inf (β = (r r β + r ( θ l + αθ l θ l + αθ l ξ sup (β = (r 4 r 3 β + r 4 ( θ l + αθ l θ l + αθ l. Therefore, when β (0, ( θ l + αθ l /, we have ξ inf (β + ξ sup (β = (r + r 3 r r 4 β + ( θ l + αθ l (r + r 4 θ l + αθ l. On the other hand, when β (( θ l + αθ l /,, we have the following computational result ξ inf (β + ξ sup (β = (r + r 3 r r 4 β + ( α(r + r 3 θ l + (r + r 4. + θ l αθ l Hence, the mean value of ξ is computed by E[ξ = ( θ l +αθ l 0 (ξ inf (β + ξ sup (β dβ + = r + r + r 3 + r 4 r r r 3 + r 4 ( αθ l. 4 The proof of theorem is complete. θ l +αθ l (ξ inf (β + ξ sup (β dβ Eample 5.4. Let ξ be the type- trapezoidal fuzzy variable defined in Eample 4.5. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 5.3, the mean value of ξ is computed by α, if α (0, 0.5 E[ξ = α, if α (0.5,. Theorem 5.5. Let ξ = ñ(µ, σ ; θ l, θ r be a type- normal fuzzy variable. Then the mean value of the CVaR reduced fuzzy variable ξ is equal to µ. Proof. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.6. Note that µ ξ (µ σ ln = µ ξ (µ + σ ln = ( θ l + αθ l /. If β (0, ( θ l + αθ l /, then ξ inf (β and ξ sup (β are the solutions of the following equation, ( µ ( θ l + αθ l ep ( σ = β. As a consequence, we have the following solutions ξ inf (β = µ σ β ln θ l + αθ l and ξ sup (β = µ + σ β ln. θ l + αθ l

17 CVaR Reduced Fuzzy Variables and Their Second Order Moments 6 Therefore, when β (0, ( θ l + αθ l /, we have ξ inf (β + ξ sup (β = µ. On the other hand, when β (( θ l + αθ l /,, we have the result ξ inf (β + ξ sup (β = µ. Hence, the mean value of ξ is computed by E[ξ = =µ. ( θ l +αθ l 0 The proof of theorem is complete. (ξ inf (β + ξ sup (β dβ + (ξ inf (β + ξ sup (β dβ θ l +αθ l Eample 5.6. Let ξ be the type- normal fuzzy variable defined in Eample 4.. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 5.5, the mean value of ξ is E[ξ =. Theorem 5.7. Let ξ = γ(λ, r; θ l, θ r be a type- gamma fuzzy variable. Suppose that, R + satisfy ( r ( ep r = ( λr λ, r ( ep r = λr λ. (i If α (0, 0.5, then the mean value of the CVaR reduced fuzzy variable ξ has the following parametric possibility distribution E[ξ = λr λr! r r r ep(r + λ r! r n (r n! r ep(r + λ ( αθ l + λr λr! r r ( r! λ n (r n! n + λn n } r n. (ii If α (0.5,, then the mean value of the CVaR reduced fuzzy variable ξ has the following parametric possibility distribution E[ξ = λr λr! r r r ep(r + λ r! r n (r n! r ep(r + λ ( αθ r + λr λr! r r ( r! λ n (r n! n + λn n } r n. Proof. We only prove the first assertion. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.9. As a consequence, the credibility distribution of ξ is Crξ } = ( θ l + αθ l ( r ( λr ep r λ, if 0 [( + θ l αθ l ( r ( λr ep r λ ( αθl, if λr [( + θ l αθ l ( r ( λr ep r λ ( αθl, if λr ( θ l + αθ l ( r ( λr ep r λ, if,

18 6 X. J. Bai and Y. K. Liu where, R + satisfy ( r ( ep r = ( λr λ, r ( ep r = λr λ. Therefore, the mean value of ξ is computed as follows E[ξ = 0 Crξ }d + + Crξ }d + λr ( αθ l + λr + λr λr! r r The proof of assertion (i is complete. Crξ }d Crξ }d = λr λr! r r ep(r + λ r ep(r + λ r ( r! λ n (r n! n + λn n r! r n (r n! } r n. Eample 5.. Let ξ be the type- gamma fuzzy variable defined in Eample 4.. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 5.7, the mean value of ξ is computed by α, if α (0, 0.5 E[ξ = α, if α (0.5,. 6. The Second Order Moments of CVaR Reduced Fuzzy Variables Let ξ be a reduced fuzzy variable with a parametric possibility distribution. To measure the variation of the parametric possibility distribution about its mean value, we adopt the following nth moment of ξ, M n [ξ = ( E[ξ n d(crξ }, (,+ where the credibility distribution is defined by the parametric possibility distribution of ξ, Crξ } = ( sup t R µ ξ (t; θ, α + sup t µ ξ (t; θ, α sup µ ξ (t; θ, α. t> When n =, M [ξ is called the second order moment of ξ. In the following, we will establish the analytical epressions of second order moments for common reduced fuzzy variables. Firstly, we have the following results about type- triangular fuzzy variable. Theorem 6.. Let ξ = ( r, r, r 3 ; θ l, θ r be a type- triangular fuzzy variable, and ξ its CVaR reduced fuzzy variable. (i If α (0, 0.5, then the second order moment of ξ is M [ξ = 4 (5r + 4r + 5r 3 4r r 6r r 3 4r r 3 6 (r r 3 ( αθ l 64 (r r + r 3 ( α θ l,

19 CVaR Reduced Fuzzy Variables and Their Second Order Moments 63 which is equivalent to the following parametric matri form M [ξ = rt P r, where r = (r, r, r 3 T, and the elements of the symmetric matri P are P = P 33 = ( α θ l 3 P = P 3 = ( α θ l 6 P 3 = ( α θ l 3 ( αθ l, + ( αθ l, + 5 4, P = ( α θl + 6. (ii If α (0.5,, then the second order moment of ξ is M [ξ = 4 (5r + 4r + 5r3 4r r 6r r 3 4r r 3 6 (r r 3 ( αθ r 64 (r r + r 3 ( α θ r, which is equivalent to the following parametric matri form M [ξ = rt P r, where r = (r, r, r 3 T, and the elements of the symmetric matri P are P = P 33 = ( α θ r 3 P = P 3 = ( α θ r 6 P 3 = ( α θ r 3 ( αθ r, + ( αθ r, + 5 4, P = ( α θr + 6. Moreover, the second order moment M [ξ is parametric quadratic conve function with respect to vector r R 3. Proof. We only prove the first assertion. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.. As a consequence, the credibility distribution of ξ is computed by Crξ } = 0, if < r ( θ l +αθ l ( r (r r, if [r, r+r (+θ l αθ l ( αθ l r r (r r, if [ r+r, r (+θ l αθ l +( αθ l r +r 3 (r 3 r, if [r, r+r3 ( θ l+αθ l (r 3 (r 3 r, if [ r+r3, r 3, if > r 3,

20 64 X. J. Bai and Y. K. Liu and the mean value of ξ is E[ξ = r + r + r 3 4 r r + r 3 ( αθ l. Therefore, the second order moment of ξ is calculated as follows M [ξ = ( E[ξ dcrξ } = θ l + αθ l (r r + + θ l αθ l (r 3 r (,+ r +r r r +r 3 r ( E[ξ d + + θ l αθ l (r r ( E[ξ d + θ l + αθ l (r 3 r r r +r r3 r +r 3 ( E[ξ d ( E[ξ d = 4 (5r + 4r + 5r 3 4r r 6r r 3 4r r 3 64 (r r + r 3 ( α θ l 6 (r r 3 ( αθ l = rt P r. On the other hand, the integrand ( E[ξ and the credibility distribution Crξ } are both nonnegative, so M [ξ 0 holds for any vector r R 3. In addition, P is a 3 3 symmetric parametric matri. Therefore, M [ξ is a positive semidefinite quadratic form. In other words, for any parameters θ l, θ r and α, the second order moment M [ξ is a parametric quadratic conve function with respect to vector r R 3. The proof of the assertion is complete. Eample 6.. Let ξ be the type- triangular fuzzy variable defined in Eample 4.3. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 6., the second order moment of ξ is computed by α, if α (0, 0.5 M [ξ = α, if α (0.5,. For the reduced fuzzy variable of a type- trapezoidal fuzzy variable, we obtain the analytical epression of second order moment in the following theorem. Theorem 6.3. Let ξ = ( r, r, r 3, r 4 ; θ l, θ r be a type- trapezoidal fuzzy variable, and ξ be its CVaR reduced fuzzy variable. M [ξ = (i If α (0, 0.5, then the second order moment of ξ is 4 (5r + 5r + 5r3 + 5r4 + r r 6r r 3 6r r 4 6r r 3 6r r 4 + r 3 r 4 ( αθ l (r r r3 + r4 r r 4 + r r 3 ( α θl (r r r 3 + r 4, 6 64 which is equivalent to the following parametric matri form M [ξ = rt Q r,

21 CVaR Reduced Fuzzy Variables and Their Second Order Moments 65 M [ξ = where r = (r, r, r 3, r 4 T, and the elements of the symmetric matri Q are Q = Q 44 = ( α θ l 3 Q = Q 34 = ( α θ l 3 Q 3 = Q 4 = ( α θ l 3 Q 4 = ( α θ l 3 Q = Q 33 = ( α θ l 3 Q 3 = ( α θ l 3 ( αθ l + 4,, + ( αθ l ( αθ l, + ( αθ l , + 5 4, (ii If α (0.5,, then the second order moment of ξ is 4 (5r + 5r + 5r 3 + 5r 4 + r r 6r r 3 6r r 4 6r r 3 6r r 4 + r 3 r 4 ( αθ r 6 (r r r3 + r4 r r 4 + r r 3 ( α θr (r r r 3 + r 4, 64 which is equivalent to the following parametric matri form M [ξ = rt Q r, where r = (r, r, r 3, r 4 T, and the elements of the symmetric matri Q are Q = Q 44 = ( α θ r 3 Q = Q 34 = ( α θ r 3 Q 3 = Q 4 = ( α θ r 3 Q 4 = ( α θ r 3 Q = Q 33 = ( α θ r 3 Q 3 = ( α θ r 3 ( αθ r + 4,, + ( αθ r ( αθ r, + ( αθ r , + 5 4, Moreover, the second order moment M [ξ is a parametric quadratic conve function with respect to vector r R 4.

22 66 X. J. Bai and Y. K. Liu Proof. We only prove the first assertion. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.4. As a consequence, the credibility distribution of ξ is calculated by Crξ } = and the mean value of ξ is 0, if < r ( θ l +αθ l ( r (r r, if [r, r+r (+θ l αθ l ( αθ l r r (r r, if [ r+r, r, if [r, r 3 (+θ l αθ l +( αθ l r 3+r 4 (r 4 r 3, if [r 3, r3+r4 ( θ l+αθ l (r 4 (r 4 r 3, if [ r3+r4, r 4, if > r 4, E[ξ = r + r + r 3 + r 4 r r r 3 + r 4 ( αθ l. 4 Therefore, the second order moment of ξ is computed as follows M [ξ = ( E[ξ dcrξ } = θ l + αθ l (r r + + θ l αθ l (r 4 r 3 (,+ r +r r r 3 +r 4 r 3 ( E[ξ d + + θ l αθ l (r r ( E[ξ d + θ l + αθ l (r 4 r 3 r r +r r4 r 3 +r 4 ( E[ξ d ( E[ξ d = 4 (5r + 5r + 5r 3 + 5r 4 + r r 6r r 3 6r r 4 6r r 3 6r r 4 + r 3 r 4 6 (r r r 3 + r 4 r r 4 + r r 3 ( αθ l 64 (r r r 3 + r 4 ( α θ l = rt Q r. On the other hand, the integrand ( E[ξ and the credibility distribution Crξ } are both nonnegative, so M [ξ 0 holds for any vector r R 4. In addition, Q is a 4 4 symmetric parametric matri. Therefore, M [ξ is a positive semidefinite quadratic form. In other words, for any parameters θ l, θ r and α, the second moment M [ξ is a parametric quadratic conve function with respect to vector r R 4. The proof of the assertion is complete. Eample 6.4. Let ξ be the type- trapezoidal fuzzy variable defined in Eample 4.5. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 6.3, the second order moment of ξ is computed by α 0.04α M [ξ =, if α (0, α 0.04α, if α (0.5,. For the reduced fuzzy variable of a type- normal fuzzy variable, the following theorem obtains the analytical epression of second order moment.

23 CVaR Reduced Fuzzy Variables and Their Second Order Moments 67 Theorem 6.5. Let ξ = ñ(µ, σ ; θ l, θ r be a type- normal fuzzy variable, and ξ be its CVaR reduced fuzzy variable. Then the second order moment of ξ is σ M [ξ = ( αθ l σ ln, if α (0, 0.5 σ ( αθ r σ ln, if α (0.5,. Moreover, the second order moment M [ξ is a parametric quadratic conve function on R with respect to σ. Proof. We only prove the first equation in the case α (0, 0.5. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.6. As a consequence, the credibility distribution of ξ is computed by Crξ } = ( θ l + αθ l ep ( ( µ σ, if µ σ ln [( + θ l αθ l ep ( ( µ σ ( αθl, if µ σ ln µ [( + θ l αθ l ep ( ( µ σ ( αθl, if µ µ + σ ln ( θ l + αθ l ep ( ( µ σ, if µ + σ ln, and the mean value of ξ is E[ξ = µ. Therefore, the second moment of ξ is calculated by M [ξ = ( µ dcrξ } (,+ = (, µ σ ln ( µ dcrξ } + (µ σ ln, µ ( µ dcrξ } + (µ, µ+σ ln ( µ dcrξ } + (µ+σ ln, + ( µ dcrξ } =σ ( αθ l σ ln. Moreover, the coefficient σ ( αθ l σ ln > 0 for any θ l, α [0,, so M [ξ 0 holds for any vector σ > 0. Thus, the second order moment M [ξ is a parametric quadratic conve function on R with respect to σ for any θ l and α. The proof of the assertion is complete. Eample 6.6. Let ξ be the type- normal fuzzy variable defined in Eample 4.. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 6.5, the second order moment of ξ is computed by 0.5( α ln, if α (0, 0.5 M [ξ = 0.35( α ln, if α (0.5,. For the reduced fuzzy variable of a type- gamma fuzzy variable, we obtain the analytical formula of the second order moment in the net theorem. Theorem 6.7. Let ξ = γ(λ, r; θ l, θ r be a type- gamma fuzzy variable, and ξ be its CVaR reduced fuzzy variable. Suppose that, R + satisfy ( r ( ep r = ( λr λ, r ( ep r = λr λ.

24 6 X. J. Bai and Y. K. Liu (i If α (0, 0.5, then the second order moment of ξ is M [ξ = (λr E[ξ λr! r r (λr + λ r+ E[ξer + λ (r +! r r n (r n +! r E[ξr! λ r n (r n! ( αθ l + λ r ( + λre[ξ λr! ( r+ λr + λ E[ξ ep(r + λ r r λe[ξ r r! ( λ n (r n! n + λn n r n (r +! (λn (r n +! n } (ii If α (0.5,, then the second order moment of ξ is. + λn n r n M [ξ = (λr E[ξ λr! r r (λr + λ r+ E[ξer + λ (r +! r r n (r n +! r E[ξr! λ r n (r n! ( αθ r + λ r ( + λre[ξ λr! ( r+ λr + λ E[ξ ep(r + λ r r λe[ξ r r! ( λ n (r n! n + λn n r n (r +! (λn (r n +! n }. + λn n r n Proof. We only prove assertion (i. Since ξ is the CVaR reduced fuzzy variable of ξ, its parametric possibility distribution µ ξ (; θ, α is given in Theorem 4.9. As a consequence, the credibility distribution of ξ is calculated by Crξ } = ( θ l + αθ l ( r ( λr ep r λ, if 0 [( + θ l αθ l ( r ( λr ep r λ ( αθl, if λr [( + θ l αθ l ( r ( λr ep r λ ( αθl, if λr ( θ l + αθ l ( r ( λr ep r λ, if, where, R + satisfy ( r ( ep r = ( λr λ, r ( ep r = λr λ. The mean value E[ξ of ξ is calculated in Theorem 5.7. Therefore, the second order moment of ξ is calculated by M [ξ = ( E[ξ dcrξ } (,+ = ( E[ξ dcrξ } + ( E[ξ dcrξ } (0, (, λr

25 CVaR Reduced Fuzzy Variables and Their Second Order Moments 69 + ( E[ξ dcrξ } + ( E[ξ dcrξ } (λr, (, + =(λr E[ξ λr! r+ ( λr + λ E[ξ ep(r + λ (r +! r r r r n (r n +! λ ( αθ l r+ +λ r E[ξr! r n (r n! + ( + λre[ξ λr! r r ( λr + λ E[ξ ep(r λ r (r +! ( λ n (r n +! n + λn n r n The proof of assertion (i is complete. λe[ξ r } r! ( λ n (r n! n + λn n r n. Eample 6.. Let ξ be the type- gamma fuzzy variable defined in Eample 4.. Suppose ξ is the CVaR reduced fuzzy variable of ξ. According to Theorem 6.7, the second order moment of ξ is computed by α α M [ξ =, if α (0, α.9α, if α (0.5,. In this eample, since the reduced fuzzy variable ξ has variable possibility distribution with respect to parameter α instead of fied possibility distribution, its second order moment also depends on the parameter α. According to above formula, we can calculate the second order moment of ξ for any given α (0,. For instance, if α = 0.5, then the second order moment is M [ξ = = A Mean-moment Optimization Model In this section, we present an application eample about portfolio selection problem, which was first proposed by Markowitz [ in stochastic environment. Different from the eisting method in the literature, we construct a new mean-moment optimization model for fuzzy portfolio selection problems, in which the mean value and second order moment discussed in the above sections are used as two important optimization indees. Given a collection of potential investments indeed from to n, let ξ i denote the return in the net time period on investment i, i =,,..., n. In the current development, we assume that ξ i is characterized by a type- fuzzy variable with known secondary possibility distribution. A portfolio is determined by specifying what fraction of one s assets to put into each investment. That is, a portfolio is a collection of nonnegative numbers i, i =,,..., n such that n i= i =. As a consequence, the return one would obtain using a given portfolio is determined by equation (. n i ξi = ξ T ( i= where ξ = ( ξ, ξ,..., ξ n T and = (,,..., n T. Given a portfolio, ξ T is also a type- fuzzy variable, and we denote its reduced fuzzy variable as r( ξ T.

26 70 X. J. Bai and Y. K. Liu One the basis of epectation criterion, the reward associated with such a portfolio can be defined as the epected return determined by equation (. [ ( n [ E r i ξi = E r( ξ T ( i= If the investor only concerns the reward, then it is simple for him to put all his assets in the investment with the highest epected return. However, it is known that investments with high rewards usually result in a high level of risk. Therefore, there is a need to define a risk measure for fuzzy return r( ξ T. In this section, we employ the second order moment to define the risk associated with the investment by equation (3. ( n M [r i ξi = M [r( ξ T (3 i= which is a quadratic deviation from the mean value E[r( ξ T. Using the notations above, we net build a new mean-moment optimization model (4 for fuzzy portfolio selection problems. ( n min M [r i ξi s.t. [ ( n E r n i = i= i= i ξi ψ i= i 0 i =,,..., n When ξ i = (r i,, r i,, r i,3, r i,4 ; θ i,l, θ i,r, i =,,..., n, are mutually independent type- trapezoidal fuzzy returns. In the case of α 0.5, according to Theorem 6.3, the second order moment of fuzzy return r( ξ T is computed by M [r( ξ T = T H, where = (,,..., n T, H = R T Q R, and the coefficient matri defined by equation (5 is the knowledge about security returns. r r r n R = r r r n r 3 r 3 r n3 (5 r 4 r 4 r n4 In the case of α > 0.5, the second order moment of fuzzy return r( ξ T is calculated by M [r( ξ T = T H where H = R T Q R, and R is determined by equation (5. (4

27 CVaR Reduced Fuzzy Variables and Their Second Order Moments 7 We net consider the equivalent parametric form of E[r( ξ T. In the case of α 0.5, according to Theorem 5.3, the mean value of fuzzy return r( ξ T is computed by [ E r( ξ T = C T, where = (,,..., n T, C = (c,, c,,..., c,n T, c,i = r i, + r i, + r i,3 + r i,4 r i, r i, r i,3 + r i,4 ( αθ l, 4 and θ l = ma i n θ i,l. In the case of α > 0.5, the mean value of fuzzy return r( ξ T is calculated by [ E r( ξ T = C T, where C = (c,, c,,..., c,n T, and c,i = r i, + r i, + r i,3 + r i,4 r i, r i, r i,3 + r i,4 ( αθ r, 4 and θ r = min i n θ i,r. As a consequence, in the case of α 0.5, model (4 is equivalent to the following parametric programming model (6. min T H s.t. C T ψ n i = i= i 0 i =,,..., n In the case of α > 0.5, model (4 is equivalent to the following parametric programming model (7. min T H s.t. C T ψ n i = i= i 0 i =,,..., n Eample 7.. Consider an investor intends to invest his fund in twenty-two securities. Let i be the investment proportion to security i, and ξ i s mutually independent type- trapezoidal fuzzy returns for i =,,...,. The parametric distributions of ξ i, i =,,..., are represented as follows. ξ = ( , ,.00,.006; θ,l, θ,r, ξ = (.00,.000,.006,.009; θ,l, θ,r, ξ 3 = ( 0.996,.0073,.00,.0094; θ 3,l, θ 3,r, ξ 4 = ( 0.993,.0096,.0,.063; θ 4,l, θ 4,r, ξ 5 = (.0033,.0,.06,.030; θ 5,l, θ 5,r, (6 (7

28 7 X. J. Bai and Y. K. Liu ξ 6 = (.046,.059,.04,.0499; θ 6,l, θ 6,r, ξ 7 = (.009,.05,.046,.0553; θ 7,l, θ 7,r, ξ = (.09,.099,.046,.0679; θ,l, θ,r, ξ 9 = (.059,.046,.06,.0709; θ 9,l, θ 9,r, ξ 0 = (.0350,.054,.067,.030; θ 0,l, θ 0,r, ξ = (.03,.0469,.070,.05; θ,l, θ,r, ξ = (.035,.069,.075,.096; θ,l, θ,r, ξ 3 = (.044,.0569,.0770,.04; θ 3,l, θ 3,r, ξ 4 = (.05,.059,.0769,.6; θ 4,l, θ 4,r, ξ 5 = (.04,.0766,.077,.6; θ 5,l, θ 5,r, ξ 6 = (.0373,.094,.097,.7; θ 6,l, θ 6,r, ξ 7 = (.0460,.093,.04,.69; θ 7,l, θ 7,r, ξ = (.0640,.0760,.30,.300; θ,l, θ,r, ξ 9 = (.065,.075,.55,.75; θ 9,l, θ 9,r, ξ 0 = (.0456,.096,.,.57; θ 0,l, θ 0,r, ξ = (.0549,.096,.79,.93; θ,l, θ,r, ξ = (.069,.099,.57,.533; θ,l, θ,r. This problem was considered in the literature, but the parameters θ i,l and θ i,r included in the secondary distributions are the same for different fuzzy returns ξ i, i =,,...,. In this section, we etend this problem by assuming that the parameters θ i,l and θ i,r are different and take on their values in the interval [0,. We build this portfolio selection problem as model (4. In this case, the portfolio selection problem is equivalent to the following parametric quadratic conve programming model (. min T H j s.t. Cj T n ψ i = i= i 0 i =,,..., where H j = R T Q j R, j =,, the matri R is defined by equation (5, and the parametric matri Q j about θ l and θ r is given in Theorem 6.3. We net solve the conve programming model ( by Lingo software. In our numerical eperiments, we set the parameters (θ,l, θ,l,..., θ,l = (

29 CVaR Reduced Fuzzy Variables and Their Second Order Moments 73 (0.05, 0.334, , , 0.649, , 0.054, 0.49, 0.357, 0.595, , 0.340, , 0.9, 0.7, 0.03, 0.046, 0.35, 0.37, , 0.6, 0.95, and (θ,r, θ,r,..., θ,r = (0.47, 0.75, 0.634, 0.75, , 0.576, 0.957, 0.003, 0.4, 0.79, , 0.49, 0.677, 0.743, , , 0.769, 0.97, 0.694, 0.950, 0.437, Thus, we have θ l = ma i n θ i,l = , and θ r = min i n θ i,r = In the case of α = 0.449, for various values of ψ, we report the obtained optimal allocation proportions to securities in Table. ψ Table. The Proportions to the Securities with α = In the case of α = 0.3, for different values of ψ, we report the obtained optimal allocation proportions to securities in Table. ψ Table. The Proportions to the Securities with α = 0.3 From Tables and, we observe that model (4 can provide diversification investments to securities. If we use a fied value or 0.3 of parameter α and change the values of parameter ψ, then the invested securities are changed accordingly. On the other hand, if we use a fied value of ψ such as.09 and take the values of α from the set 0.449, 0.3}, then the invested securities are changed from securities 3 and 0 to securities, 3 and. Even if the invested securities are the same, the invested proportions to them are often different. As

30 74 X. J. Bai and Y. K. Liu a consequence, the computational results demonstrate that the invested securities and their invested proportions in our portfolio selection problem depend heavily on the possibility distributions of fuzzy returns. By the definition of parameter α, it determines the possibility distributions of fuzzy returns. In summary, the computational results demonstrate that parametric possibility distributions have some advantages over fied possibility distributions when we employ them to model fuzzy portfolio selection problems.. Conclusions and Future Research In this paper, we presented a new reduction method for type- fuzzy variables, and obtained the following major new results. (i We defined the CVaR for regular fuzzy variable, and established some useful CVaR formulas for common regular fuzzy variables. (ii We proposed the CVaR reduction method for type- fuzzy variables. For the reduced triangular, trapezoidal, normal and gamma fuzzy variables, we derived their useful parametric possibility distributions. (iii According to the parametric possibility distributions of the reduced triangular, trapezoidal, normal and gamma fuzzy variables, we established some useful analytical formulas of mean values. (iv For reduced fuzzy variables, the nth moments were first defined to gauge the variations of parametric possibility distributions with respect to their mean values. Then, we established some useful analytical formulas of second order moments for common reduced fuzzy variables. (v Applying the second order moment as a new risk measure, we developed a mean-moment optimization method for fuzzy portfolio selection problems. The solution results reported in the numerical eperiments demonstrated that parametric possibility distributions have some advantages over fied possibility distributions when we employ them to model fuzzy portfolio selection problems. Future research might address the following two aspects. First, this paper established some useful analytical epressions about mean values and second order moments of common reduced fuzzy variables. The analytical epressions about higher order moments of reduced fuzzy variables and their sums should be studied in the near future. Second, as far as the practical applications are concerned, this paper suggested to use the second order moment of reduced fuzzy variable as a measure to gauge the risk resulted from fuzzy uncertainty. The theoretical results obtained in this paper have potential applications in other practical risk management and engineering optimization problems, including transportation problem, location and allocation problem, data envelopment analysis, emergency supplies prepositioning problem and so on. Acknowledgements. The authors wish to thank Editors and anonymous referees for their valuable comments, which help us to improve the paper a lot. This work was supported by the National Natural Science Foundation of China (No , and the Training Foundation of Hebei Province Talent Engineering.