
In this article, we focus on the BV regularity of the adapted entropy solutions of the conservation laws whose flux function contains infinitely many discontinuities with possible accumulation points. It is well known that due to discontinuities of the flux function in the space variable, the total variation of the solution can blow up to infinity in finite time. We establish the existence of total variation bounds for certain classes of fluxes and the initial data. Furthermore, we construct two counterexamples, which exhibit BV blow-up of the entropy solution. These counterexamples not only demonstrate that these assumptions are essential, but also show that the BV-regularity result of [S. S. Ghoshal, J. Differential Equations, 258 (3), 980-1014, 2015] does not hold true when the spatial discontinuities of the flux are infinite.
Citation: Shyam Sundar Ghoshal, John D. Towers, Ganesh Vaidya. BV regularity of the adapted entropy solutions for conservation laws with infinitely many spatial discontinuities[J]. Networks and Heterogeneous Media, 2024, 19(1): 196-213. doi: 10.3934/nhm.2024009
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In this article, we focus on the BV regularity of the adapted entropy solutions of the conservation laws whose flux function contains infinitely many discontinuities with possible accumulation points. It is well known that due to discontinuities of the flux function in the space variable, the total variation of the solution can blow up to infinity in finite time. We establish the existence of total variation bounds for certain classes of fluxes and the initial data. Furthermore, we construct two counterexamples, which exhibit BV blow-up of the entropy solution. These counterexamples not only demonstrate that these assumptions are essential, but also show that the BV-regularity result of [S. S. Ghoshal, J. Differential Equations, 258 (3), 980-1014, 2015] does not hold true when the spatial discontinuities of the flux are infinite.
Tapezoidal intuitionistic fuzzy number (TIFN), as a special intuitionistic fuzzy set on a real number, has the best capability to model ill-known quantities [1,2]. TIFNs are useful to deal with uncertain knowledge and data in management decision and financial engineering problems. Intuitionistic fuzzy number, as the extension of fuzzy number, has more flexibility in many real-time uncertain application fields such as multi-criteria decision [3,4,5,6], portfolio selection [7,8], and agammaegation [9]. Recently, the possibility measure theory of TIFN plays an important role in many application fields like mulit-criteria decision and portfolio selection problem. For example, Chakraborty [10] utilized possibility, necessity and credibility measure to solve intuitionistic fuzzy optimization problem. Garai [11] applied possibility-necessity-credibility measures of generalized trapezoidal intuitionistic fuzzy number (GTIFN) to multi-product manufacturing system.
As is well known, based on possibility theory the possibilistic mean and variance of fuzzy numbers were presented by Carlsson and Full´er [12] and had been widely applied in solving fuzzy portfolio decision problems [13,14,15,16,17,18,19,20]. Recently, inspired by the above-mentioned fuzzy possibilistic mean theory Wan [21] proposed a definition of possibilistic mean and variance of intuitionistic fuzzy number (IFN). And Wan [22,23,24] utilized the possibilistic mean and variance to intuitionistic fuzzy group decision-making and matrix game problems with incomplete information. Later, Chen [25] also discussed the mean-variance-skewness portfolio selection problems based on intuitionistic fuzzy optimization under the assumption that the returns of assets are intuitionistic fuzzy numbers. Although the possibilistic mean and variance of intuitionistic fuzzy numbers will have a lot of application prospects, there is few study on the relationship between the possibilistic mean, variance and covariance of two generalized trapezoidal intuitionistic fuzzy numbers, and it is not clear whether subsethood entails smaller variance for generalized trapezoidal intuitionistic fuzzy number.
Moreover, we notice that the existing definitions of possibilistic mean and variance by Wan [21] have some flaws. In his definitions the denominators of the fraction of possibilistic mean and variance are improper because the level λ value and γ value are taken in [0, 1] for granted. However, the level λ value and γ level value of generalized trapezoidal intuitionistic fuzzy number ˜A indeed lie in the limited interval [0, ua] and [va, 1], respectively. To overcome the flaw of Wan's definition [21], in this paper we will correct it and introduce the new definitions of lower and upper possibilistic mean, variance of generalized trapezoidal intuitionistic fuzzy numbers based on possibility distributions. We also give some properties of possibilistic mean, covariance and variance, such as that the possibilistic variance of linear combinations of generalized trapezoidal intuitionistic fuzzy numbers can easily be computed like in probability theory and that subsethood does entail smaller variance. Moreover, in this paper we show that the possibilistic mean of generalized trapezoidal intuitionistic fuzzy number remains additive in the sense of addition of GTIFNs. In particular, we easily deduce Var(˜A)=Cov(˜A,˜A)=Cov(˜A,−˜A), which is not consistent with the properties of covariance in crisp probability theory. The contribution of this work is as follows. We show that the presented possibilistic variance and covariance of generalized trapezoidal intuitionistic fuzzy numbers reserve many important properties of variance and covariance in traditional probability theory, which is different from the known investigation results. We also investigate the important relationship between two kinds of possibilistic variance var(˜A) and var′(˜A) for generalized trapezoidal intuitionistic fuzzy numbers.
Let us first introduce some basic concepts of trapezoidal intuitionistic fuzzy numbers, which will be employed in the following sections.
Definition 1 [11,21]. A generalized trapezoidal intuitionistic fuzzy number (GTIFN) ˜A=((a,b,c,d);ua,va) is a special kind of intuitionistic fuzzy set on the real number set R, whose membership function and nonmembership function are defined as following forms, respectively (see Figure 1).
μ˜A(x)={ua(x−a)/(b−a),ifa⩽x<b,ua,ifb⩽x⩽c,ua(d−x)/(d−c),ifc<x⩽d,0,otherwise.v˜A(x)={va+(1−va)(b−x)/(b−a),ifa⩽x<b,va,ifb⩽x⩽c,va+(1−va)(x−c)/(d−c),ifc<x⩽d,1,otherwise. |
where ua,va are the greatest membership and the least nonmembership degree, respectively. And ua,va∈[0,1], 0⩽ua+va⩽1, π˜A(x)=1−μ˜A(x)−ν˜A(x) is called the hesitation degree of x∈˜A.
Remark 1 [22,24]. If b=c, then the generalized trapezoidal intuitionistic fuzzy number ˜A=((a,b,c,d);ua,va) is reduced to a generalized triangular intuitionistic fuzzy number ˜A=((a,b,d);ua,va).
Definition 2. Let ˜Ai=((ai,bi,ci,di),ui,vi), i=1,2, be two generalized trapezoidal intuitionistic fuzzy numbers, the addition and scale multiplication of GTIFNs [2,11,21,24,26] are defined as follows.
(1) ˜A1+˜A2=((a1+a2,b1+b2,c1+c2,d1+d2);ua1∧ua2,va1∨va2).
(2) x˜Ai=((xai,xbi,xci,xdi);ui,vi), ∀x⩾0. x˜Ai=((xdi,xci,xbi,xai);ui,vi), ∀x<0.
Definition 3 [22,24]. Let ˜A=((a,b,c,d);ua,va) be a generalized trapezoidal intuitionistic fuzzy number, the λ-level cut set and γ -cut set of membership and nonmembership of GTIFN ˜A are, respectively, defined as
˜A[λ]={x/μ˜A(x)⩾λ}=[a−(λ),a+(λ)]=[a+(b−a)λ/ua,d−(d−c)λ/ua)]; |
˜A[γ]={x/ν˜A(x)⩽γ}=[a−(γ),a+(γ)]=[(b−ava−γ(b−a))/(1−va),(c−dva+γ(d−c))/(1−va))]. |
where λ∈[0,ua], γ∈[va,1], and 0⩽λ+γ⩽1.
Theorem 1 [11,21]. Let ˜A1=((a1,b1,c1,d1);ua1,va1) , ˜A2=((a2,b2,c2,d2);ua2,va2) be two generalized trapezoidal intuitionistic fuzzy numbers, for any positive real number λ∈(0,ua], γ∈[va,1] , we can easily get
(1) (˜A1+˜A2)[λ]=˜A1[λ]+˜A2[λ]=[a1−(λ)+a2−(λ),a1+(λ)+a2+(λ)], ∀λ∈(0,ua];
(2) (˜A1+˜A2)[γ]=˜A1[γ]+˜A2[γ]=[a1−(γ)+a2−(γ),a1+(γ)+a2+(γ)], ∀γ∈[va,1];
(3) (x˜A)[λ]=x˜A[λ]=[xa−(λ),xa+(λ)],∀x⩾0; (x˜A)[λ]=x˜A[λ]=[xa+(λ),xa−(λ)],∀x<0;
(4) (x˜A)[γ]=x˜A[γ]=[xa−(γ),xa+(γ)],∀x⩾0; (x˜A)[γ]=x˜A[γ]=[xa+(γ),xa−(γ)],∀x<0.
The proof is straightforward from Definition 3 and Zadeh's fuzzy extension principle.
Definition 4 [21]. Let ˜A=((a,b,c,d);ua,va) be a generalized trapezoidal intuitionistic fuzzy number, the lower and upper possibilistic mean value of the membership function of GTIFN ˜A are, respectively, defined as
M−μ(˜A)=∫ua0pos(μ˜A⩽a−(λ))a−(λ)dλ∫ua0pos(μ˜A⩽a−(λ))dλ=∫ua0λa−(λ)dλ∫ua0λdλ=2u2a∫ua0λa−(λ)dλ=13(a+2b); |
M+μ(˜A)=∫ua0pos(μ˜A⩾a+(λ))a+(λ)dλ∫ua0pos(μ˜A⩾a+(λ))dλ=∫ua0λa+(λ)dλ∫ua0λdλ=2u2a∫ua0λa+(λ)dλ=13(d+2c). |
The possibilistic mean of membership of GTIFN ˜A can also be defined as
Mμ(˜A)=[M−μ(˜A)+M+μ(˜A)]2==1u2a∫ua0λ(a−(λ)+a+(λ))dλ=16(a+2b+2c+d). |
Definition 5. ˜A=((a,b,c,d);ua,va) be a generalized trapezoidal intuitionistic fuzzy number, the lower and upper possibilistic mean value of the nonmembership function of GTIFN ˜A are, respectively, defined as
M−ν(˜A)=∫1vapos(b⩾ν˜A⩾a−(γ))a−(γ)dγ∫1vapos(b⩾ν˜A⩾a−(γ))dγ=∫1vaγa−(γ)dγ∫1vaγdγ=21−v2a∫1vaγa−(γ)dγ |
=11−v2a[(b−ava)(1+va)−23(b−a)(1+va+v2a)]; |
M+ν(˜A)=∫1vapos(c⩽ν˜A⩽a+(γ))a+(γ)dγ∫1vapos(c⩽ν˜A⩽a+(γ))dγ=∫1vaγa+(γ)dγ∫1vaγdγ=21−v2a∫1vaγa+(γ)dγ |
=11−v2a[(c−dva)(1+va)+23(d−c)(1+va+v2a)]. |
The possibilistic mean of nonmembership of GTIFN ˜A can also be defined as
Mν(˜A)=[M−ν(˜A)+M+ν(˜A)]2=11−v2a∫1vaγ(a−(γ)+a+(γ))dγ=11−v2a[12(b+c−(a+d)va)(1+va)+13(a−b−c+d)(1+va+v2a)]. |
Definition 6. Let ˜A=((a,b,c,d);ua,va) be a generalized trapezoidal intuitionistic fuzzy number, the possibilistic mean value of GTIFN ˜A is defined as M(˜A)=12[Mμ(˜A)+Mν(˜A)].
Definition 7. Let ˜A=((a,b,c,d);ua,va) be a GTIFN with λ,γ cut set ˜A[λ]=[a−(λ),a+(λ)], ˜A[γ]=[a−(γ),a+(γ)] of membership and nonmembership of ˜A, the possibilistic variance of membership and nonmembership of GTIFN ˜A are respectively defined as [21,22],
varμ(˜A)=12∫ua0(a−(λ)−a+(λ))2λdλ=124u2a[(d−a)2−2(d−a)(b−c)+3(b−c)2]; |
varν(˜A)=12∫1va(a−(γ)−a+(γ))2γdγ=124(1−va)[6(1+va)M2+8(1+va+v2a)MN+3(1+va+v2a+v3a)N2], |
where M=c−b+(a−d)va, N=d−c+b−a.
Hence, the variance of GTIFN ˜A can be defined by var(˜A)=12[varμ(˜A)+varv(˜A)].
Definition 8. Let ˜A1=((a1,b1,c1,d1);ua,va) be a GTIFN with λ,γ cut set ˜A1[λ]=[a−1(λ),a+1(λ)], ˜A1[γ]=[a−1(γ),a+1(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua,va) with λ,γ cut set ˜A2[λ]=[a−2(λ),a+2(λ)], ˜A2[γ]=[a−2(γ),a+2(γ)] of membership and nonmembership, the possibilistic covariance of membership and nonmembership of GTIFNs ˜A1,˜A2 are respectively defined as [17,19],
covμ(˜A1,˜A2)=12∫ua0(a+1(λ)−a−1(λ))(a+2(λ)−a−2(λ))λdλ; |
covν(˜A1,˜A2)=12∫1va(a+1(γ)−a−1(γ))(a+2(γ)−a−2(γ))γdγ. |
The covariance of GTIFNs ˜A1,˜A2 can be also defined by cov(˜A1,˜A2)=12[covμ(˜A1,˜A2)+covv(˜A1,˜A2)].
Definition 9. ˜A=((a,b,c,d);ua,va) be a GTIFN with λ cut set ˜A[λ]=[a−(λ),a+(λ)] of membership and γ cut set ˜A[γ]=[a−(γ),a+(γ)] of nonmembership, the new possibilistic variance of membership and nonmembership of GTIFN ˜A are respectively defined as
var′μ(˜A)=∫ua0[(Mμ(˜A)−a−(λ))2+(Mμ(˜A)−a+(λ))2]λdλ; |
var′ν(˜A)=∫1va[(Mv(A)−a−(γ))2+(Mv(A)−a+(γ))2]γdγ. |
The new possibilistic variance of GTIFN ˜A can also be computed by var′(˜A)=12[var′μ(˜A)+var′v(˜A)]; where a−(λ),a+(λ) is the lower bound and upper bound of λ level interval value of membership of GTIFN ˜A, var′μ(˜A) expresses the weighted deviation squares between the membership mean and lower bound, upper bound of λ level interval of membership of ˜A; and a−(γ),a+(γ) is the lower bound and upper bound of λ level interval of nonmembership of GTIFN ˜A, var′ν(˜A) expresses the weighted deviation squares between the nonmembership mean and lower bound, upper bound of λ level interval of nonmembership of ˜A. However, the existing varμ(˜A) only indicates the weighted deviation squares between lower bound and upper bound of λ level interval of membership of ˜A; varv(˜A) only indicates the weighted deviation squares between lower bound and upper bound of λ level interval of nonmembership of ˜A. Moreover, the above-mentioned definition 9 of possibilistic variance coincides with the definition of crisp variance, so it is more meaningful and rathional than the existing possibilistic variance of GTIFN ˜A.
Definition 10. Let ˜A1=((a1,b1,c1,d1);ua,va) be a GTIFN with λ,γ cut set ˜A1[λ]=[a−1(λ),a+1(λ)] , ˜A1[γ]=[a−1(γ),a+1(γ)] and TIFN ˜A2=((a2,b2,c2,d2);ua,va) with λ,γ cut set ˜A2[λ]=[a−2(λ),a+2(λ)] , ˜A2[γ]=[a−2(γ),a+2(γ)] of membership and nonmembership, the new possibilistic covariance of membership and nonmembership of TIFNs ˜A1,˜A2 are respectively defined as
cov′μ(˜A1,˜A2)=∫ua0[(Mμ(˜A1)−a1−(λ))(Mμ(˜A2)−a−2(λ))+(Mμ(˜A1)−a1+(λ))(Mμ(˜A2)−a2+(λ))]λdλ;cov′ν(˜A1,˜A2)=∫1va[(Mν(˜A1)−a1−(γ))(Mν(˜A2)−a−2(γ))+(Mν(˜A1)−a1+(γ))(Mν(˜A2)−a2+(γ))]γdγ. |
The new covariance of GTIFNs ˜A1,˜A2 can be computed by cov′(˜A1,˜A2)=12[cov′μ(˜A1,˜A2)+cov′v(˜A1,˜A2)].
Theorem 2. Let
(1) M(˜A1+˜A2)=M(˜A1)+M(˜A2);
(2) M(x˜A1)=xM(˜A1), ∀x∈R;
(3) M(x1˜A1+x2˜A2)=x1M(˜A1)+x2M(˜A2), ∀x1,x2∈R.
Proof. (1) It follows from Definitions 4, 5 and Theorem 1 that
Mμ(˜A1+˜A2)=1u2a∫ua0λ[(a−1(λ)+a−2(λ))+(a+1(λ)+a+2(λ))]dλ=1u2a∫ua0λ[(a−1(λ)+a+1(λ))+(a−2(λ)+a+2(λ))]dλ |
=1u2a[∫ua0λ(a−1(λ)+a+1(λ))dλ+∫ua0λ(a−2(λ)+a+2(λ))dλ] |
=Mμ(˜A1)+Mμ(˜A2); |
Mv(˜A1+˜A2)=11−v2a∫1vaγ[(a−1(γ)+a−2(γ))+(a+1(γ)+a+2(γ))]dγ=11−v2a∫1vaγ[(a−1(γ)+a+1(γ))+(a−2(γ)+a+2(γ))]dγ |
=11−v2a[∫1vaγ(a−1(γ)+a+1(γ))dλ+∫1vaγ(a−2(γ)+a+2(γ))dγ] |
=Mv(˜A1)+Mv(˜A2). |
Therefore, we have
M(˜A1+˜A2)=Mμ(˜A1+˜A2)+Mv(˜A1+˜A2)2=M(˜A1)+M(˜A2). |
(2) By Definitions 4, 5 and Theorem 1, for any x≥0 we easily get
Mμ(x˜A1)=1u2a∫ua0[x(a−1(λ)+a+1(λ))]λdλ=x[1u2a∫ua0λ(a−1(λ)+a+1(λ))dλ]=xMμ(˜A1); |
Mv(x˜A1)=11−v2a∫1va[x(a−1(γ)+a+1(γ))]γdγ=x[11−v2a∫1vaγ(a−1(γ)+a+1(γ))dγ]=xMv(˜A1). |
For any x<0, we also get
Mμ(x˜A1)=1u2a∫ua0[x(a+1(λ)+a−1(λ))]λdλ=x[1u2a∫ua0λ(a+1(λ)+a−1(λ))dλ]=xMμ(˜A1); |
Mv(x˜A1)=11−v2a∫1va[x(a+1(γ)+a−1(γ))]γdγ=x[11−v2a∫1vaγ(a+1(γ)+a−1(γ))dγ]=xMv(˜A1). |
So we have
M(x˜A1)=Mμ(x˜A1)+Mv(x˜A1)2=xMμ(˜A1)+xMv(˜A1)2=xM(˜A1). |
(3) From (1), (2) we easily conclude that, for any x1,x2∈R,
Mμ(x1˜A1+x2˜A2)=Mμ(x1˜A1)+Mμ(x2˜A2)=x1Mμ(˜A1)+x2Mμ(˜A2); |
Mv(x1˜A1+x2˜A2)=Mv(x1˜A1)+Mv(x2˜A2)=x1Mv(˜A1)+x2Mv(˜A2). |
Hence, M(x1˜A1+x2˜A2)= Mμ(x1˜A1+x2˜A2)+Mv(x1˜A1+x2˜A2)2=x1M(˜A1)+x2M(˜A2).
Theorem 3.1. Let ˜A1=((a1,b1,c1,d1);ua1,va1) be a GTIFN with λ,γ cut set ˜A1[λ]=[a−1(λ),a+1(λ)] , ˜A1[γ]=[a−1(γ),a+1(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua2,va2) with λ,γ cut set ˜A2[λ]=[a−2(λ),a+2(λ)] , ˜A2[γ]=[a−2(γ),a+2(γ)] and μa1=μa2=μa,va1=va2=va , and let constant θ∈R . Then we have
(1) cov(˜A1,˜A2)=cov(˜A2,˜A1);
(2) var(˜A1)=cov(˜A1,˜A1)=cov(˜A1,−˜A1);
(3) var(˜A1+θ)=var(˜A1).
Proof. (1), (2) can be directly verified by Definition 8 of cov(˜A1,˜A2) and Definition 7 of var(˜A1).
(3) For any θ∈R, by Zadeh's fuzzy extension principle we can know that (˜A1+θ)(x)=˜A1(x−θ), and
(˜A1+θ)[λ]=˜A[λ]1+θ=[a−1(λ)+θ,a+1(λ)+θ]; (˜A1+θ)[γ]=˜A[γ]1+θ=[a−1(γ)+θ,a+1(γ)+θ]. |
So, varμ(˜A1+θ)=12∫ua0[(a1−(λ)+θ)−(a1+(λ)+θ)]2λdλ=12∫ua0[a1−(λ)−a1+(λ)]2λdλ=varμ(˜A1);
varv(˜A1+θ)=12∫1va[(a1−(γ)+θ)−(a1+(λ)+θ)]2γdγ=12∫1va[a1−(γ)−a1+(λ)]2γdγ=varv(˜A1).
Hence, we have var(˜A1+θ)=varμ(˜A1+θ)+varv(˜A1+θ)2=varμ(˜A1)+varv(˜A1)2=var(˜A1).
Theorem 3.2. Let ˜A1=((a1,b1,c1,d1);ua1,va1) be a GTIFN with λ,γ cut set ˜A1[λ]=[a1−(λ),a1+(λ)], ˜A1[γ]=[a1−(γ),a1+(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua2,va2) with λ,γ cut set ˜A2[λ]=[a2−(λ),a2+(λ)], ˜A2[γ]=[a2−(γ),a2+(γ)], and μa1=μa2=μa,va1=va2=va, and for any constant θ∈R, we also obtain
(1) cov′(˜A1,˜A2)=cov′(˜A2,˜A1);
(2) var′(˜A1)=cov′(˜A1,˜A1);
(3) var′(˜A1+θ)=var′(˜A1).
Proof. (1), (2) can be directly proved by Definition 10 of cov′(˜A1,˜A2) and Definition 9 of var′(˜A1).
(3) Since (˜A1+θ)(x)=˜A1(x−θ), we know that
(˜A1+θ)[λ]=˜A[λ]1+θ=[a−1(λ)+θ,a+1(λ)+θ]; (˜A1+θ)[γ]=˜A[γ]1+θ=[a−1(γ)+θ,a+1(γ)+θ]. |
So, var′μ(˜A1+θ)=∫ua0[(Mμ(˜A1+θ)−(a1−(λ)+θ))2+(Mμ(˜A1+θ)−(a1+(λ)+θ))2]λdλ |
=∫ua0[(Mμ(˜A1)+θ−a1−(λ)−θ)2+(Mμ(˜A1)+θ−a1+(λ)−θ)2]λdλ=∫ua0[(Mμ(˜A1)−a1−(λ))2+(Mμ(˜A1)−a1+(λ))2]λdλ=var′(˜A1); |
var′ν(˜A1+θ)=∫1va[(Mv(˜A1+θ)−(a1−(γ)+θ))2+(Mv(˜A1+θ)−(a1+(γ)+θ))2]γdγ=∫1va[(Mv(˜A1)−a1−(γ))2+(Mv(˜A1)−a1+(γ))2]γdγ=var′ν(˜A1); |
So, we get var′(˜A1+θ)=var′(˜A1).
Theorem 3.1-3.2 show that possibilistic covariances have symmetry, the covariance between GTIFN itself reduces to the possibilistic variance of GTIFN. And if a GTIFN shifts a constant, the possibilistic variance is not altered.
Theorem 3.3. Let ˜A1=((a1,b1,c1,d1);ua1,va1) be a GTIFN with λ,γ cut set ˜A1[λ]=[a1−(λ),a1+(λ)] , ˜A1[γ]=[a1−(γ),a1+(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua2,va2) with λ,γ cut set ˜A2[λ]=[a2−(λ),a2+(λ)] , ˜A2[γ]=[a2−(γ),a2+(γ)] , and μa1=μa2=μa,va1=va2=va , for any x1,x2∈R , then
var(x1˜A1+x2˜A2)=x12var(˜A1)+x22var(˜A2)+2|x1x2|cov(˜A1,˜A2) |
Proof. We only verify that the above formula holds in the case (1) if x1<0, x2>0.
By employing Theorem 1 we can easily know that
(x1˜A1+x2˜A2)[λ]=[x1a+1(λ)+x2a−2(λ),x1a−1(λ)+x2a+2(λ)]; |
(x1˜A1+x2˜A2)[γ]=[x1a+1(γ)+x2a−2(γ),x1a−1(γ)+x2a+2(γ)]; |
So, by Definitions 7, 8 we easily get
varμ(x1˜A1+x2˜A2) |
=12∫ua0λ[(x1a+1(λ)+x2a−2(λ))−(x1a−1(λ)+x2a+2(λ))]2dλ |
=12∫ua0λ[x1(a+1(λ)−a−1(λ))−x2(a+2(λ)−a−2(λ))]2dλ=12{∫ua0λx21(a+1(λ)−a−1(λ))2dλ+∫ua0λx22(a+2(λ)−a−2(λ))2dλ−∫ua02λx1x2(a+1(λ)−a−1(λ))(a+2(λ)−a−2(λ))dλ}=x12varμ(˜A1)+x22varμ(˜A2)−2x1x2covμ(˜A1,˜A2) |
=x12varμ(˜A1)+2|x1x2|covμ(˜A1,˜A2)+x22varμ(˜A2); |
and
varv(x1˜A1+x2˜A2) |
=12∫1vaγ[(x1a+1(γ)+x2a−2(γ))−(x1a1−(γ)+x2a+2(γ))]2dγ=12{∫1vaγx21(a+1(γ)−a−1(γ))2dγ+∫1vaγx22(a−2(γ)−a+2(γ))2dγ−∫1va2γx1x2(a+1(γ)−a−1(γ))(a+2(γ)−a−2(γ))dγ}=x12varv(˜A1)+x22varv(˜A2)−2x1x2covv(˜A1,˜A2) |
=x12varv(˜A1)+2|x1x2|covv(˜A1,˜A2)+x22varv(˜A2). |
So, we get var(x1˜A1+x2˜A2)=varμ(x1˜A1+x2˜A2)+varν(x1˜A1+x2˜A2)2
=x12var(˜A1)+2|x1x2|cov(˜A1,˜A2)+x22var(˜A2). |
Similar reasoning holds for the other cases (2) x1>0, x2>0, (3) x1>0, x2<0 and (4) x1⩽0, x2⩽0.
It follows that var(x1˜A1+x2˜A2)=x12var(˜A1)+x22var(˜A2)+2|x1x2|cov(˜A1,˜A2) for any x1,x2∈R.
Theorem 3.3 show that the possibilistic variance of linear combinations of generalized trapezoidal intuitionistic fuzzy numbers can be computed like in probability theory, which reserves the same property of crisp variance.
Theorem 3.4. Let ˜A1=((a1,b1,c1,d1);ua1,va1) be a GTIFN with λ,γ cut set ˜A1[λ]=[a1−(λ),a1+(λ)] , ˜A1[γ]=[a1−(γ),a1+(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua2,va2) with λ,γ cut set ˜A2[λ]=[a2−(λ),a2+(λ)] , ˜A2[γ]=[a2−(γ),a2+(γ)] , and μa1=μa2=μa,va1=va2=va , for any x,x1,x2∈R , then
var′(x1˜A1+x2˜A2)=x12var′(˜A1)+x22var′(˜A2)+2|x1x2|cov′(φ(x1)˜A1,φ(x2)˜A2), |
where φ(x) is a signal function of x∈R.
Proof. We only need to prove it in the case (1) when x1>0, x2<0.
By using Theorem 1 we easily have
(x1˜A1+x2˜A2)[λ]=[x1a−1(λ)+x2a+2(λ),x1a+1(λ)+x2a−2(λ)]; |
(x1˜A1+x2˜A2)[γ]=[x1a−1(γ)+x2a+2(γ),x1a+1(γ)+x2a−2(γ)]. |
With Definitions 9, 10 and Theorem 2 we get
var′μ(x1˜A1+x2˜A2)=∫ua0[Mμ(x1˜A1+x2˜A2)−(x1a1−(λ)+x2a2+(λ))]2λdλ+[Mμ(x1˜A1+x2˜A2)−(x1a1+(λ)+x2a2−(λ))]2λdλ=∫ua0{[x1Mμ(˜A1)+x2Mμ(˜A2)−x1a1−(λ)−x2a2+(λ)]2+[x1Mμ(˜A1)+x2Mμ(˜A2)−x1a1+(λ)−x2a2−(λ)]2}λdλ=∫ua0{[x1(Mμ(˜A1)−a1−(λ))+x2(Mμ(˜A2)−a2+(λ))]2+[x1(Mμ(˜A1)−a1+(λ))+x2(Mμ(˜A2)−a2−(λ))]2}λdλ=x12var′μ(˜A1)+2x1x2∫ua0λ[(M(˜A1)−a1−(λ))(Mμ(˜A2)−a2+(λ))+(Mμ(˜A1)−a1+(λ))(Mμ(˜A2)−a2−(λ))]dλ+x22var′μ(˜A2)=x12var′μ(˜A1)−2x1x2∫ua0λ[(M(˜A1)−a1−(λ))(Mμ(−˜A2)+a2+(λ))+(Mμ(˜A1)−a1+(λ))(Mμ(−˜A2)+a2−(λ)))]dλ+x22var′μ(˜A2)=x12var′μ(˜A1)−2x1x2cov′μ(˜A1,−˜A2)+x22var′μ(˜A2)=x12var′μ(˜A1)+2|x1x2|cov′μ(φ(x1)˜A1,φ(x2)˜A2)+x22var′μ(˜A2); |
By Definitions 9, 10 and Theorem 2 we also get
var′v(x1˜A1+x2˜A2)=∫1va[Mv(x1˜A1+x2˜A2)−(x1a1−(γ)+x2a2+(γ)]2γdγ+[Mv(x1˜A1+x2˜A2)−(x1a1+(γ)+x2a2−(γ))]2γdγ=∫1va[x1Mv(˜A1)+x2Mv(˜A2)−x1a1−(γ)−x2a2+(γ)]2+[x1Mv(˜A1)+x2Mv(˜A2)−x1a1+(γ)−x2a2−(γ)]2}γdγ=∫1va[x1(Mv(˜A1)−a1−(γ))+x2(Mv(˜A2)−a2+(γ))]2+[x1(Mv(˜A1)−a1+(γ))+x2(Mv(˜A2)−a2−(γ))]2}γdγ=x12var′v(˜A1)−2x1x2∫1vaγ[(Mv(˜A1)−a1−(γ))(Mv(−˜A2)+a2+(γ))+(Mv(˜A1)−a1+(γ))(Mv(−˜A2)+a2−(γ))]dγ+x22var′v(˜A2)=x12var′v(˜A1)−2x1x2cov′v(˜A1,−˜A2)+x22var′v(˜A2)=x12var′v(˜A1)+2|x1x2|cov′v(φ(x1)˜A1,φ(x2)˜A2)+x22var′v(˜A2). |
Hence, we verify that
var′(x1˜A1+x2˜A2)=12[var′μ(x1˜A1+x2˜A2)+var′v(x1˜A1+x2˜A2)] |
=x12var′(˜A1)+2|x1x2|cov′(φ(x1)˜A1,φ(x2)˜A2)+x22var′(˜A2). |
Similar reasoning holds for the other cases (2) x1>0 , x2>0; (3) x1<0 , x2>0 and (4) x1⩽0, x2⩽0, which ends the proof.
Theorem 3.4. shows that the new possibilistic variance of linear combinations of generalized trapezoidal intuitionistic fuzzy numbers can also be computed like in probability theory, which reserves the similar property of crisp variance in traditional probability space.
Theorem 3.5. Let ˜A1=((a1,b1,c1,d1);ua1,va1) be a GTIFN with λ,γ cut set ˜A1[λ]=[a1−(λ),a1+(λ)] , ˜A1[γ[=[a1−(γ),a1+(γ)] and GTIFN ˜A2=((a2,b2,c2,d2);ua2,va2) with λ,γ cut set ˜A2[λ]=[a2−(λ),a2+(λ)] , ˜A2[γ[=[a2−(γ),a2+(γ)] , μa1=μa2=μa,va1=va2=va , and ˜A1⊂˜A2 . Then we have
(1) var(˜A1)⩽var(˜A2), (2) var′(˜A1)⩽var′(˜A2).
Proof.
(1) From ˜A1⊂˜A2 it follows that a−2(λ)⩽a−1(λ)⩽a+1(λ)⩽a+2(λ) and a−2(γ)⩽a−1(γ)⩽a+1(γ)⩽a+2(γ).
For all λ∈(0,ua), γ∈(va,1). That is to say, M−μ(˜A2)⩽M−μ(˜A1)⩽M+μ(˜A1)⩽M+μ(˜A2).
Hence, varμ(˜A1)=12∫ua0[a+1(λ)−a−1(λ)]2λdλ⩽12∫ua0[a+2(λ)−a−2(λ)]2λdλ=varμ(˜A2),
varv(˜A1)=12∫1va[a+1(γ)−a−1(γ)]2γdγ⩽12∫1va[a+2(γ)−a−2(γ)]2γdγ=varv(˜A2).
So, we have
var(˜A1)=varμ(˜A1)+varν(˜A1)2⩽varμ(˜A2)+varν(˜A2)2=var(˜A2). |
(2) From Definition 9, we know that
var′μ(˜A2)−var′μ(˜A1)=∫ua0{[Mμ(˜A2)−a2−(λ)]2+[Mμ(˜A2)−a2+(λ)]2}λdλ−∫ua0{[Mμ(˜A1)−a1−(λ)]2+[Mμ(˜A1)−a1+(λ)]2}λdλ=∫ua0[Mμ(˜A2)−Mμ(˜A1)+a1−(λ)−a2−(λ)][Mμ(˜A2)+Mμ(˜A1)−a1−(λ)−a2−(λ)]λdλ+∫ua0[Mμ(˜A2)−Mμ(˜A1)+a1+(λ)−a2+(λ)][Mμ(˜A2)+Mμ(˜A1)−a1+(λ)−a2+(λ)]λdλ |
By Definition 4 and Tchebycheff inequalities we infer that
var′μ(˜A2)−var′μ(˜A1)⩾∫ua0[Mμ(˜A2)−Mμ(˜A1)+a1−(λ)−a2−(λ)]λdλ×∫ua0[Mμ(˜A2)+Mμ(˜A1)−a1−(λ)−a2−(λ)]λdλ+∫ua0[Mμ(˜A2)−Mμ(˜A1)+a1+(λ)−a2+(λ)]λdλ×∫ua0[Mμ(˜A2)+Mμ(˜A1)−a1+(λ)−a2+(λ)]λdλ=(u2a2)2[Mμ(˜A2)−Mμ(˜A1)+M−μ(˜A1)−M−μ(˜A2)]×[Mμ(˜A2)+Mμ(˜A1)−M−μ(˜A1)−M−μ(˜A2)]+(u2a2)2[Mμ(˜A2)−Mμ(˜A1)+M+μ(A1)−M+μ(A2)]×[Mμ(˜A2)+Mμ(˜A1)−M+μ(A1)−M+μ(A2)]=u4a2[(M+μ(˜A2)−M−μ(˜A2)2)2−(M+μ(A1)−M−μ(A1)2)2]⩾0 |
Similarly we can verify that var′ν(˜A2)−var′ν(˜A1)⩾0.
So, we have var′(˜A2)=var′μ(˜A2)+var′ν(˜A2)2⩾var′μ(˜A1)+var′ν(˜A1)2=var′(˜A1).
The above Theorem 3.5 in fact indicates that the subsethood of generalized trapezoidal intuitionistic fuzzy number does entail smaller variance.
Theorem 3.6. Let ˜A=((a,b,c,d);ua,va) be a generalized trapezoidal intuitionistic fuzzy number with λ,γ cut set ˜A[λ]=[a−(λ),a+(λ)], ˜A[γ]=[a−(γ),a+(γ)] of the corresponding membership and nonmembership of ˜A, respectively, and 0⩽ua+va⩽1, then we obtain var(˜A)⩽var′(˜A).
Proof. From Definitions 7, 9, 4 we easily know that
varμ(˜A)=12∫ua0[(a+(λ))2+(a−(λ))2]λdλ−∫ua0a+(λ)a−(λ)λdλ |
var′μ(˜A)=∫ua0{[Mμ(˜A)−a−(λ)]2+[Mμ(˜A)−a+(λ)]2}λdλ=∫ua0[(a+(λ))2+(a−(λ))2]λdλ−∫ua02(Mμ(˜A))[a−(λ)+a+(λ)]λdλ+2∫ua0(Mμ(˜A))2λdλ=∫ua0[(a+(λ))2+(a−(λ))2]λdλ−2(Mμ(˜A))2u2a+(Mμ(˜A))2u2a. |
From the Jensen inequality and ∫ua02λu2adλ=1, we get
[∫ua02λu2a(a−(λ)+a+(λ))dλ]2⩽∫ua02λu2a(a−(λ)+a+(λ))2dλ. |
That is,
4u4a[∫ua0λ(a−(λ)+a+(λ))dλ]2⩽ 2u2a∫ua0λ(a−(λ)+a+(λ))2dλ.
i.e., 1u2a[∫ua0λ(a−(λ)+a+(λ))dλ]2⩽12 ∫ua0λ(a−(λ)+a+(λ))2dλ.
So, varμ(˜A)−var′μ(˜A)
=−12∫ua0[(a+(λ))2+(a−(λ))2]λdλ−∫ua0a+(λ)a−(λ)λdλ+2(Mμ(˜A))2u2a−(Mμ(˜A))2u2a=−12∫ua0[(a+(λ))2+(a−(λ))2]λdλ−∫ua0a+(λ)a−(λ)λdλ+(Mμ(˜A))2u2a=−12∫ua0[a+(λ)+a−(λ)]2λdλ+1u2a[∫ua0λ(a−(λ)+a+(λ))dλ]2⩽0. |
Similarly, by Definitions 7, 9, 5 we can prove
varν(˜A)=12∫1va[(a+(γ))2+(a−(γ))2]γdγ−∫1vaa+(γ)a−(γ)γdγ;var′ν(˜A)=∫1va{[Mν(˜A)−a−(γ)]2+[Mν(˜A)−a+(γ)]2}γdγ=∫1va[(a+(γ))2+(a−(γ))2]γdγ−∫1va2(Mν(˜A))[a−(γ)+a+(γ)]γdγ+2∫1va(Mν(˜A))2γdγ=∫1va[(a+(γ))2+(a−(γ))2]γdγ−2(Mν(˜A))2(1−v2a)+(Mν(˜A))2(1−v2a). |
From the Jensen inequality and ∫1va2γ1−v2adγ=1, we get
[∫1va2γ1−v2a(a−(γ)+a+(γ))dγ]2⩽∫1va2γ1−v2a(a−(γ)+a+(γ))2dγ. |
That is, 4(1−v2a)2[∫1vaγ(a−(γ)+a+(γ))dγ]2⩽ 2(1−v2a)∫1vaγ(a−(γ)+a+(γ))2dγ.
i.e., 1(1−v2a)[∫1vaγ(a−(γ)+a+(γ))dγ]2⩽ 12∫1vaγ(a−(γ)+a+(γ))2dγ.
So, varν(˜A)−var′ν(˜A)
=−12∫1va[(a+(γ))2+(a−(γ))2]γdγ−∫1vaa+(γ)a−(γ)γdγ+2(Mv(˜A))2(1−v2a)−(Mv(˜A))2(1−v2a)=−12∫1va[(a+(λ))2+(a−(λ))2]λdλ−∫1vaa+(λ)a−(λ)λdλ+(Mν(˜A))2(1−v2a)=−12∫1va[a+(λ)+a−(λ)]2λdλ+11−v2a[∫1vaγ(a−(γ)+a+(γ))dγ]2⩽0. |
Therefore, we get var(˜A)=varμ(˜A)+varν(˜A)2⩽var′μ(˜A)+var′ν(˜A)2=var′(˜A).
The above Theorem 3.6 indicates that the new presented possibilistic variance var′(˜A) of generalized trapezoidal intuitionistic fuzzy number is greater than the existing variance var(˜A) because it considers all the weighted deviation squares between the lower bound, upper bound of level cut interval of membership of ˜A and possibilistic mean.
Remark 2. If ˜A=((a,b,c,d);ua,va) is a symmetric generalized trapezoidal intuitionistic fuzzy number, ua,va are two constants satisfying 0⩽ua+va⩽1, then we have var(˜A)=var′(˜A).
Proof. Since ˜A=((a,b,c,d);ua,va) is a symmetric TIFN, we know a+d=b+c and d−c=b−a.
For any λ∈[0,ua], γ∈[va,1], the λ-level cut set and γ -cut set of TIFN ˜A can be expressed as follows.
˜A[λ]={x/μ˜A(x)⩾λ}=[a−(λ),a+(λ)]=[a+(b−a)λ/ua,d−(d−c)λ/ua)] |
=[b−s(λ),c+s(λ)]; |
˜A[γ]={x/ν˜A(x)⩽γ}=[a−(γ),a+(γ)]=[(b−ava−γ(b−a))/(1−va),(c−dva+γ(d−c))/(1−va)]. |
=[b−q(γ),c+q(γ)]. |
where s(λ)=b−a−(b−a)λ/ua, q(γ)=[(a−b)va+γ(b−a)]/(1−va).
According to Definitions 4, 5 of possibilistic mean of TIFN ˜A, one easily get
M(˜A)=1u2a∫ua0λ[a−(λ)+a+(λ)]dλ=1u2a∫ua0λ[b−s(λ)+c+s(λ)]dλ=12(b+c),Mν(˜A)=11−v2a∫1vaγ[a−(γ)+a+(γ)]dγ=11−v2a∫1vaγ[b−q(γ)+c+q(γ)]dγ=12(b+c). |
Hence,
varμ(˜A)=12∫ua0[a−(λ)−a+(λ)]2λdλ=12∫ua0[(b−s(λ))−(c+s(λ))]2λdλ=12∫ua0[b−c−2s(λ)]2λdλ;varν(˜A)=12∫1va[a−(γ)−a+(γ)]2γdγ=12∫1va[(b−q(γ))−(c+q(γ))]2γdγ=12∫1va[b−c−2q(γ)]2γdγ. |
And var′μ(˜A)=∫ua0[(Mμ(˜A)−a−(λ))2+(Mμ(˜A)−a+(λ))2]λdλ
=∫ua0λ([b+c2−(b−s(λ))]2+[b+c2−(c+s(λ))]2)dλ |
=∫ua0λ([c−b2+s(λ)]2+[b−c2−s(λ)]2)dλ=∫ua02λ([b−c2−s(λ)]2)dλ=12∫ua0λ[b−c−2s(λ)]2dλ=varμ(˜A); |
var′ν(˜A)=∫1va[(Mv(˜A)−a−(γ))2+(Mv(˜A)−a+(γ))2]γdγ |
=∫1vaγ([Mv(˜A)−(b−q(γ))]2+[Mv(˜A)−(c+q(γ)]2)dγ |
=∫1vaγ([b+c2−(b−q(γ))]2+[b+c2−(c+q(γ))]2)dγ=∫1vaγ([c−b2+q(γ))]2+[b−c2−q(γ)]2)dγ=12∫1vaγ[b−c−2q(γ)]2dγ=varν(˜A). |
Thus, var(˜A)=varμ(˜A)+varν(˜A)2=var′μ(˜A)+var′ν(˜A)2=var′(˜A).
Recently, some arithmetic operations of generalized trapezoidal intuitionistic fuzzy numbers have been utilized in analyzing fuzzy system reliability [26,27]. Also, many multi-attribute decision making and programming methods based on generalized trapezoidal intuitionistic fuzzy number have been studied in [28,29,30]. Although GTIFNs have been applied in dealing with uncertain problem, there is few study on the possibilistic mean and variance of GTIFN. In this section we will discuss some advantages of the proposed possibilistic mean and variance of GTIFN below.
In fact, for any GTIFN ˜A=((a,b,c,d);ua,va), the possibilistic mean M(˜A) may be respectively viewed as the central value of GTIFN ˜A, and the possibilistic variance Var(˜A) may be regarded as the global spreads of GTIFN ˜A. Thus, from the membership function point of view, the larger the possibilistic mean, the greater the GTIFN; the smaller the possibilistic variance, the greater the GTIFN.
Hence, we can define the rank index of generalized trapezoidal intuitionistic fuzzy number as
Rank(˜A)=M(˜A)−ρVar(˜A), |
where ρ represents the risk attitude of decision-maker(DM). If 0⩽ρ<0.5, it shows that DM is risk-like; If ρ=0.5, it shows that DM is risk-neutral; If 0.5<ρ⩽1, it shows that DM is risk-averse.
The new ranking method of GTIFNs {˜Ai} can be obtained according to the ascending order of rank index as following:
(1) If Rank(˜Ai)<Rank(˜Aj) then ˜Ai<˜Aj;
(2) If Rank(˜Ai)=Rank(˜Aj) then ˜Ai=˜Aj;
(3) If Rank(˜Ai)>Rank(˜Aj) then ˜Ai>˜Aj.
In this part, we discuss the portfolio decision problem with generalized trapezoidal intuitionistic fuzzy number returns. Let us consider a fuzzy portfolio selection problem. Suppose investor has initial wealth 1 and may buy m assets in the capital investment. The investor intends to allocate his/her wealth among the m risky assets for making accounting investment plan. X=(x1,x2,⋯,xm) is the investor portfolio vector representing the capital allocation vector among m securities, xi is the investment capital proportion of risky asset i; li, ui are the lower bound and upper bound of the capital invested in risky asset i(1⩽i⩽m), respectively. Suppose the future return rates of the alternative m assets are easily assessed by GTIFN ˜Ai=((ai,bi,ci,di);uai,vai), ∀i=1,2,⋯,m. In order to construct the intuitionistic fuzzy portfolio decision model, we first introduce two important concepts which will be used in the proposed inutitionistic fuzzy portfolio decision model.
By using the previous Theorem 2 and Definition 6 we can compute the expected mean of fuzzy return of portfolio X=(x1,x2,⋯,xm) as E(X)=∑mj=1xiM(˜Ai). By using the above-mentioned Theorem 3.4 and Definition 9 we can evaluate the possibilistic variance of intuitionistic fuzzy return of portfolio X=(x1,x2,⋯,xm) as
Var(X)=Var(∑mi=1xi˜Ai)=∑mi=1Var(˜Ai)+2∑1⩽i<j<mxixjcov(˜Ai,˜Aj). |
Assume that the whole investment process is self-financing, that is, the investor does not invest the additional capital during the portfolio selection. Inspired by the classic Markovitz portfolio theory, we establish the following intuitionistic fuzzy portfolio model by maximizing the expected mean of portfolio return and minimizing the possibilistic variance risk of portfolio as below.
maxE(X)=∑mi=1xiM(˜Ai)s.t.{Var(∑mi=1xi˜Ai)≤κ∑mi=1xi=10≤li≤xi≤ui≤1, |
where κ is the selected parameters by investor reflecting the risk attitude of investor. It means the upper bound of risk tolerance of the portfolio movement.
By applying Lingo or Matlab Optimization Toolbox we can easily solve the above programming models. Finally, we can obtain the optimal portfolio strategy X∗=(x∗1,x∗2,⋯,x∗m), which is the optimal solver corresponding to the maximum objective function value, x∗i is the optimal investment wealth proportion in risky asset i (1⩽i⩽m) under the risk constraints.
In this article, we introduce some notes of possibilistic variance of generalized trapezoidal intuitionistic fuzzy number and then discuss some important properties about possibilistic mean and variance of GTIFN based on possibility theory. The presented possibilistic mean and variance of GTIFN can be used to measure the expected return and risk of portfolio with GTIFN return. In the future, we will investigate the intuitionistic fuzzy portfolio decision-making and selection problem by maximizing possibilistic mean and minimizing possibilistic variance of portfolio.
We thank the Editor and the anonymous referees for their good suggestions and comments. This paper is supported by the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2018A030313996.
The authors declare no conflict of interest in this paper.
[1] |
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