Calculation formulas and correlation inequalities for variance bounds and semi-variances of fuzzy intervals

Abstract

Variance in fuzzy set theory, generally applied in investment decision, risk evaluation, and so on, can be described as a measurement that gauges the deviation of a fuzzy number. In this paper, in order to extend the application range and enrich the research area of variance, the concepts of variance bounds and semi-variances are defined and discussed from a theoretical point of view. With respect to some frequently-used fuzzy intervals, four relatively simple calculation formulas for upper and lower bounds of variance, and upside and downside semi-variances are put forward respectively, with the aid of which several correlation inequalities are subsequently presented and proved. Besides, in order to depict the concepts and inequalities more distinctly, plenty of examples are introduced to make some numerical illustration.

Keywords

Fuzzy interval bounds of variance semi-variance inequalities

1. Introduction

Variance, as one of the significant conceptions in probability theory, has been verified to be a predominant instrument in the measurement of the deviation of a random variable or a set of data from its expected value, with the assistance of which diverse complex problems can be modelled and solved with preferable results especially risk assessment [2 , 15]. Thus, in view of the admirable performance of variance in probability theory, Liu [12] in 2002 initially extended variance to fuzzy set theory to gauge the deviation of a fuzzy number, the propose of which not only promotes the development of fuzzy set theory but also assists people in managing the intricate problems in real applications. From a theoretical point of view, on the basis of variance, subsequent work further dived into it and then a set of vital definitions in fuzzy set theory were put forward involving covariance [6], central moment [13], etc. Besides, in real applications, various practical auxiliary tools, such as non-Shannon fuzzy mean-variance-skewness-entropy model [18] and fuzzy quantity mean-variance view [1], were developed by taking advantage of the underlying philosophy of variance.

Despite of the extensive applications both in theory and practice, fuzzy variance still has distinguished limitation in applications. One is the complicated calculation procedures of variance as pointed out by Yi et al. [19]. The calculation of variance defined in [12] is always with quantities of complex processes, which immensely increase the difficulty of its acquisition such that the variance can only be used restrictively. Meanwhile, with respect to many practical problems such as portfolio, in which high returns are what the decision-makers prefer, the utilization of variance may lead to too much sacrifice of the expected high returns and thus give rise to the produce of unexpected results since high positive and low negative deviations are undesirable equally according to Huang [8].

To improve the weakness of variance for more effective utilization, further studies of variance have been increasingly carried out in subsequential work. In view of the complexity of the calculation, an upper bound of variance, together with a simple calculation formula for some commonly-used fuzzy numbers, was proposed by Yi et al. [19], which commendably decreases the complexity and then makes the extension of applications of variance possible in the practical problems. Moreover, Huang [8] introduced the concept of semi-variance into fuzzy set theory to manage the situations with undesirability of downside deviation, which may assist investors in the choice of the portfolio with high returns. Following that, various concepts and definitions related to variance were raised as well for improving its existing boundedness. For instance, on the basis of fuzzy variance, Li et al. [10] further proposed the concept of skewness, which can be applied to multiassets portfolio selection problem to help investors in the acquirement of the optimal investment decision under complex market situations. Besides, other researches concerning the extension of fuzzy variance were also made (see, e.g., [11, 20]). By now, fuzzy variance and its derivatives have been widely applied to diverse areas under fuzzy environments [4, 16].

As a further investigation on fuzzy variance from the theoretical perspective and a succeeding work of the above mentioned literature as well, in this paper, a lower bound of the variance and the upside semi-variance are initiated, which may be applied to the situations that variance is adept at or upside deviations are undesirable in, respectively. Moreover, on the basis of the two new concepts, together with the upper bound of the variance in [19] and downside semi-variance in [8], four calculation formulas with low complexity are then proposed in the light of the recently presented operational law in Zhou et al. [26]. Subsequently, some inequalities related to the relationships among these concepts are proved together with quite a few numerical examples for illustration.

The rest of this paper is organized as follows. In Section 2, an overall review on the basic concepts of fuzzy interval is given. The concept of fuzzy variance is introduced in Section 3, together with the calculation results of some specific fuzzy intervals. Based on the concepts of upper bound of the variance and downside semi-variance, Sections 4 and 5 discuss the lower bound of the variance and upside semi-variance, respectively, and further derive the simplified computation formulas. Section 6 describes and verifies several inequalities about the relationships among these related concepts. Finally, the conclusions are drawn in Section 7.

2. Preliminaries

Serving as a significant measurement in fuzzy set theory, variance shows its superiority on the depiction of the degree of the dispersion of a fuzzy number. In this section, for further research on the properties of variance in the following sections, the basic knowledge including fuzzy interval, credibility distribution, expected value and some other relates, are reviewed consecutively.

2.1. Fuzzy interval

Fuzzy numbers have been widely used for dealing with vague information since Zadeh [21] initiated the fuzzy set theory in 1965. As a special but frequently-used type of fuzzy numbers, fuzzy interval plays a critical part in the description of some relatively complex vague information [7, 25], among which one typical instance is the trapezoidal fuzzy number.

Dubios and Prade [5] defined a fuzzy interval as a fuzzy quantity with a quasi-concave membership function μ (x), i.e., a convex fuzzy subset of the real line $ℝ$ such that $μ (z) \geq \min {μ (x), μ (y)}, \forall x, y \in ℝ, z \in [x, y] .$

Moreover, a fuzzy interval, ϱ, is of LR-type if there exist shape functions L (for left), R (for right), and four parameters: $(\underline{m}, \bar{m}) \in ℝ^{2} ⋃ {- \infty, + \infty}, γ > 0, β > 0$ with membership function $μ (x) = {\begin{matrix} L (\frac{\underline{m} - x}{γ}), & if x < \underline{m} \\ 1, & if \underline{m} \leq x \leq \bar{m} \\ R (\frac{x - \bar{m}}{β}), & if x > \bar{m} . \end{matrix}$

Please see [5 , 26] for more details about the definition of shape functions.

Example 1. With respect to a fuzzy interval ϱ, if its membership function is $μ (x) = {\begin{matrix} \frac{x - i}{\underline{m} - i}, & if i \leq x < \underline{m} \\ 1, & if \underline{m} \leq x \leq \bar{m} \\ \frac{p - x}{p - \bar{m}}, & if \bar{m} < x \leq p \\ 0, & otherwise, \end{matrix}$ which is depicted in Figure 1, then ϱ is called a trapezoidal fuzzy number and can be represented by $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ , where the four real numbers i, $\underline{m}$ , $\bar{m}$ , and p are with the relationship $i < \underline{m} < \bar{m} < p$ . Note that when $\underline{m} - i = p - \bar{m}$ , the trapezoidal fuzzy number is said to be symmetric; otherwise, it is called asymmetric.

Fig.1

The membership function of a trapezoidal fuzzy number $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ in Example 1.

Example 2. With respect to a fuzzy interval ϱ, if its membership function is $μ (x) = {\begin{matrix} e^{- (\frac{x - \underline{m}}{i})^{2}}, & if x < \underline{m} \\ 1, & if \underline{m} \leq x \leq \bar{m} \\ e^{- (\frac{x - \bar{m}}{i})^{2}}, & if x > \bar{m}, \end{matrix}$ which is depicted in Figure 2, then ϱ is called a Gaussian fuzzy interval, denoted as $ϱ \sim (i, \underline{m}, \bar{m})$ , where the real numbers $\underline{m}$ and $\bar{m}$ are with the relationship $\underline{m} < \bar{m}$ .

Fig.2

The membership function of a Gaussian fuzzy interval $ϱ \sim (i, \underline{m}, \bar{m})$ in Example 2.

2.2. Credibility distribution and inverse credibility distribution

Credibility measure was initially defined by Liu and Liu [14] as the average of possibility measure [22] and necessity measure [23], which has been proved in [14] to be self-dual and subadditive. For any fuzzy number ϱ and real number t, the credibility of the fuzzy event ϱ ≤ t can be calculated via the membership function μ of ϱ,that is, ${ϱ \leq t} = \frac{1}{2} (sup_{x \leq t} μ (x) + 1 - sup_{x > t} μ (x)) .$ (1)

Subsequently, in virtue of credibility measure, the definition of credibility distribution of a fuzzy number ϱ was defined by Liu [12] as Γ (x) = Cr {ϱ ≤ x}, $x \in ℝ$ .

Example 3. With respect to the trapezoidal fuzzy number $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ in Example 1, based on the definition of credibility distribution, we can obtain that the credibility distributionof ϱ is

$Γ (x) = {\begin{matrix} 0, & if x \leq i \\ \frac{x - i}{2 (\underline{m} - i)}, & if i < x \leq \underline{m} \\ 0.5, & if \underline{m} < x \leq \bar{m} \\ \frac{x + p - 2 \bar{m}}{2 (p - \bar{m})}, & if \bar{m} < x \leq p \\ 1, & if x > p, \end{matrix}$ (2) which is depicted in Figure 3.

Fig.3

The credibility distribution of a trapezoidal fuzzy number $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ in Example 3.

Example 4. In regard to a Gaussian fuzzy interval $ϱ \sim (i, \underline{m}, \bar{m})$ in Example 2, we can get itscredibility distribution as $Γ (x) = {\begin{matrix} \frac{1}{2} e^{- (\frac{x - \underline{m}}{i})^{2}}, & if x \leq \underline{m} \\ 0.5, & if \underline{m} < x \leq \bar{m} \\ 1 - \frac{1}{2} e^{- (\frac{x - \bar{m}}{i})^{2}}, & if x > \bar{m}, \end{matrix}$ which is depicted in Figure 4.

Fig.4

The credibility distribution of a Gaussian fuzzy interval $ϱ \sim (i, \underline{m}, \bar{m})$ in Example 4.

By comparing the aforesaid two graphical examples, several observable similarities can be immediately obtained. First of all, the credibility distributions of the two fuzzy intervals are both continuous and strictly increasing with respect to $x < \underline{m}$ or $x > \bar{m}$ . Secondly, there always exists a closed interval $[\underline{m}, \bar{m}]$ in which the credibility takes the value of 0.5. In order to describe fuzzy intervals with such properties, Zhou et al. [26] further dived into them and then proposed some practical definitions as follows.

Definition 1. (Zhou et al. [26]) An LR fuzzy interval is said to be regular if the shape functions L and R are continuous and strictly decreasing functions with respect to x at which 0 < L (x) <1 and 0 < R (x) <1, respectively.

Definition 2. (Zhou et al. [26]) Let ϱ be a regular LR fuzzy interval with credibility distribution Γ (x). Then the inverse credibility distribution of ϱ can be defined by ${\begin{matrix} sup {x | Γ (x) = β}, \\ if β = 0 \\ Γ^{- 1} (β), \\ if β \in (0, 0.5) \cup (0.5, 1) \\ [inf {x | Γ (x) = β}, sup {x | Γ (x) = β}], \\ if β = 0.5 \\ inf {x | Γ (x) = β}, \\ if β = 1, \end{matrix}$ and denoted as Γ^-1 (β) for the sake of simplicity.

Note that the inverse credibility distribution of a regular LR fuzzy interval, Γ^-1 (β), is continuous and strictly increasing on the domains (0, 0.5) and (0.5, 1).

Example 5. With respect to a trapezoidal fuzzy number $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ discussed in Examples 1 and 3, its inverse credibility distribution is $Γ^{- 1} (β) = {\begin{matrix} 2 (\underline{m} - i) β + i, & if 0 \leq β < 0.5 \\ [\underline{m}, \bar{m}], & if β = 0.5 \\ 2 (p - \bar{m}) β + 2 \bar{m} - p, & if 0.5 < β \leq 1, \end{matrix}$ (3) which is depicted in Figure 5.

Fig.5

The inverse credibility distribution of a trapezoidal fuzzy number $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ in Example 5.

Example 6. With respect to a Gaussian fuzzy interval $ϱ \sim (i, \underline{m}, \bar{m})$ discussed in Examples 2 and 4, its inverse credibility distribution is $Γ^{- 1} (β) = {\begin{matrix} \underline{m} - i \sqrt{- ln (2 β)}, & if 0 \leq β < 0.5 \\ [\underline{m}, \bar{m}], & if β = 0.5 \\ \bar{m} + i \sqrt{- ln (2 - 2 β)}, & if 0.5 < β \leq 1, \end{matrix}$ (4) which is depicted in Figure 6.

Fig.6

The inverse credibility distribution of a Gaussian fuzzy interval $ϱ \sim (i, \underline{m}, \bar{m})$ in Example 6.

Theorem 1. (Zhou et al. [26], Operational Law) Let ϱ₁, ϱ₂, ⋯ , ϱ_n be independent regular LR fuzzy intervals with credibility distributions Γ₁, Γ₂, ⋯ , Γ_n, respectively. If the function f (x₁, x₂, ⋯ , x_n) is continuous and strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then ϱ = f (ϱ₁, ⋯ , ϱ_m, ϱ_m+1, ⋯ , ϱ_n) is a regular LR fuzzy interval with inverse credibilitydistribution

\begin{matrix} Ψ^{- 1} (β) = & f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)), \end{matrix}

(5)

where $Γ_{1}^{- 1}, Γ_{2}^{- 1}, \dots, Γ_{n}^{- 1}$ are the inverse credibility distributions of ϱ₁, ϱ₂, ⋯ , ϱ_n, respectively.

2.3. Expected value

Expected value operator in fuzzy set theory acts as a bridge between the credibility measure and the variance of a fuzzy number. In virtue of credibility measure, Liu and Liu [14] presented the definition of expected value for fuzzynumbers.

Definition 3. (Liu and Liu [14]) Let ϱ be a fuzzy number. Then its expected value is defined by $E [ϱ] = \int_{0}^{+ \infty} {ϱ \geq t} d t - \int_{- \infty}^{0} {ϱ \leq t} t .$ Subsequently, with respect to a regular LR fuzzy interval, an equivalent form of expected value operator was given by Zhou et al. [26] from the perspective of inverse credibility distribution.

Theorem 2. (Zhou et al. [26]) Let ϱ be a regular LR fuzzy interval. If its expected value exists, then

E [ϱ] = \int_{0}^{1} Γ^{- 1} (β) β,

(6)

where Γ^-1 is the inverse credibility distribution of ϱ.

Example 7. Let ϱ be a trapezoidal fuzzy number $T \sim (i, \underline{m}, \bar{m}, p)$ . Then, in terms of Equations (9) and (13), we have $\begin{matrix} E [ϱ] & = \int_{0}^{0.5} (2 (\underline{m} - i) β + i) d β \\ + \int_{0.5}^{1} (2 (p - \bar{m}) β + 2 \bar{m} - p) β \\ = \frac{i + \underline{m} + \bar{m} + p}{4} . \end{matrix}$

Moreover, for the special case $\underline{m} - i = p - \bar{m}$ , which means that ϱ is a symmetric trapezoidal fuzzy number, the expected value of ϱ, E [ϱ], becomes $(\underline{m} + \bar{m}) / 2$ .

Example 8. Let ϱ be a Gaussian fuzzy interval $\sim (i, \underline{m}, \bar{m})$ . Then, in terms of Equations (10) and (13), we have $\begin{matrix} E [ϱ] & = \int_{0}^{0.5} (\underline{m} - i \sqrt{- ln (2 β)}) d β \\ + \int_{0.5}^{1} (\bar{m} + i \sqrt{- ln (2 - 2 β)}) d β \\ = \frac{\underline{m} + \bar{m}}{2} - (i - i) \int_{0}^{1} \sqrt{- ln t} d t \\ = \frac{\underline{m} + \bar{m}}{2} . \end{matrix}$

Theorem 3. (Zhou et al. [26]) Let ϱ₁, ϱ₂, ⋯ , ϱ_n be independent regular LR fuzzy intervals with credibility distributions Γ₁, Γ₂, ⋯ , Γ_n, respectively. If the continuous function f (x₁, x₂, ⋯ , x_n) is strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the expected value of regular LR fuzzy interval ϱ = f (ϱ₁, ⋯ , ϱ_m, ϱ_m+1, ⋯ , ϱ_n) is

\begin{matrix} E [ϱ] = & \int_{0}^{1} f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) β, \end{matrix}

(7)

where $Γ_{1}^{- 1}, Γ_{2}^{- 1}, \dots, Γ_{n}^{- 1}$ are the inverse credibility distributions of ϱ₁, ϱ₂, ⋯ , ϱ_n, respectively.

Theorem 4. (Liu and Liu [14]) Let ϱ and υ be independent fuzzy numbers with finite expected values. Then for any real numbers i and j, we have $E [i ϱ + j υ] = iE [ϱ] + jE [υ] .$

3. Variance

Suppose that ϱ is a fuzzy number with expected value e. From Liu [12], the variance of ϱ is defined as $V [ϱ] = E [(ϱ - e)^{2}] .$ (8)

From Definition 1, Eq. (16) can be reformed as $V [ϱ] = \int_{0}^{+ \infty} {(ϱ - e)^{2} \geq t} d t .$ (9)

Based on Eq. (17), the variance of the two kinds of fuzzy intervals mentioned above, including trapezoidal fuzzy numbers and Gaussian fuzzy intervals, can be obtained immediately.

Example 9. Let $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ be a symmetric trapezoidal fuzzy number. Then the variance of ϱ can be calculated through the following procedures. Firstly, it follows from Eq. (16) that V [ϱ] = E [(ϱ - e) ²], where $e = (\underline{m} + \bar{m}) / 2$ from Example 7. Secondly, denote υ = (ϱ - e) ², whose membership function is obtained as $μ_{υ} (x) = {\begin{matrix} 1, & if 0 \leq x < (\frac{\bar{m} - \underline{m}}{2})^{2} \\ \frac{p - i - 2 \sqrt{x}}{2 (p - \bar{m})}, & if (\frac{\bar{m} - \underline{m}}{2})^{2} \leq x \leq (\frac{p - i}{2})^{2} \\ 0, & if x > (\frac{p - i}{2})^{2}, \end{matrix}$ which is depicted in Figure 7.

Fig.7

The membership function of υ in Example 7.

Thirdly, in terms of the self-duality of Cr and Eq. (2), the expression of Cr {υ ≥ r} can be easily derived as

$\begin{matrix} {υ \geq r} = 1 - {υ < r} = \\ {\begin{matrix} \frac{1}{2}, & if 0 \leq r < (\frac{\bar{m} - \underline{m}}{2})^{2} \\ \frac{p - i - 2 \sqrt{r}}{4 (p - \bar{m})}, & if (\frac{\bar{m} - \underline{m}}{2})^{2} \leq r \leq (\frac{p - i}{2})^{2} \\ 0, & otherwise . \end{matrix} \end{matrix}$ (10)

Finally, in the light of Equations (17) and (20), the variance of ϱ is $\begin{matrix} V [ϱ] & = \int_{0}^{+ \infty} {υ \geq r} d r \\ = \frac{(\bar{m} - \underline{m})^{2}}{8} + \frac{(p - i)^{3}}{48 (p - \bar{m})} - \frac{(p - i) (\bar{m} - \underline{m})^{2}}{16 (p - \bar{m})} \\ + \frac{(\bar{m} - \underline{m})^{3}}{24 (p - \bar{m})} . \end{matrix}$ (11)

Provided that ϱ ∼ (2, 4, 8, 10), we then have V [ϱ] =14/3.

Example 10. Let $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ be an asymmetric trapezoidal fuzzy number, where $e = (i + \underline{m} + \bar{m} + p) / 4$ from Example 7. Without loss of generality, assume that $\underline{m} - i < p - \bar{m}$ .

Since the value of e is possible to be located in the interval either $(\underline{m}, \bar{m})$ or $[\bar{m}, p)$ , thus the following will discuss the two cases separately.

Case 1: $\underline{m} < e < \bar{m}$ . Assuming that υ = (ϱ - e) ², V [ϱ] can be then substituted by E [υ], and themembership function of υ is calculated as $μ_{υ} (x) = {\begin{matrix} 1, \\ if 0 \leq x < (\underline{m} - e)^{2} \\ \frac{e - i - \sqrt{x}}{\underline{m} - i}, \\ if (\underline{m} - e)^{2} \leq x \leq (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} \\ \frac{p - e - \sqrt{x}}{p - \bar{m}}, \\ if (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} < x \leq (p - e)^{2} \\ 0, \\ otherwise, \end{matrix}$ as shown in Figure 8.

Fig.8

The membership function of υ in Example 8.

From the self-duality of Cr and Eq. (2), it can be derived as $\begin{matrix} {υ \geq t} = \\ {\begin{matrix} \frac{1}{2}, \\ if 0 \leq t < (\underline{m} - e)^{2} \\ \frac{e - a - \sqrt{t}}{2 (\underline{m} - i)}, \\ if (\underline{m} - e)^{2} \leq t \leq (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} \\ \frac{p - e - \sqrt{t}}{2 (p - \bar{m})}, \\ if (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} < t \leq (p - e)^{2} \\ 0, \\ if t > (p - e)^{2} . \end{matrix} \end{matrix}$ (12)

Eventually, on the basis of Equations (17) and (22), we can get the variance of ϱ as $\begin{matrix} V [ϱ] & = \int_{0}^{+ \infty} {υ \geq t} d r \\ = \int_{0}^{(\underline{m} - e)^{2}} \frac{1}{2} d t \\ + \int_{(\underline{m} - e)^{2}}^{(\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2}} \frac{\bar{m} - e - \sqrt{t}}{2 (\underline{m} - i)} d t \\ + \int_{(\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2}}^{(p - e)^{2}} \frac{p - e - \sqrt{t}}{2 (p - \bar{m})} d t . \end{matrix}$ (13)

In view of the extreme complication of the integrals involved in Eq. (23), we employ Matlab to make the corresponding calculation, which outputs $\begin{matrix} V [ϱ] & = \frac{\bar{m} - e}{2 (\underline{m} - i)} (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} \\ + \frac{p - e}{2 (p - \bar{m})} (\frac{e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p}{p - \bar{m} + i - \underline{m}})^{2} \\ + \frac{(e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p)^{3}}{2 (p - \bar{m}) (p - \bar{m} + i - \underline{m})^{3}} \\ - \frac{(e (p - \bar{m} + \underline{m} - i) + i \bar{m} - \underline{m} p)^{3}}{3 (\underline{m} - i) (p - \bar{m} + i - \underline{m})^{3}} \\ - \frac{(\bar{m} - e) (\underline{m} - e)^{2}}{2 (\underline{m} - i)} + \frac{(3 \underline{m} - i - \bar{m} - p)^{2}}{32} \\ + \frac{(\underline{m} - e)^{3}}{3 (\underline{m} - i)} . \end{matrix}$ (14)

Provided that $ϱ \sim T (1, 2, 5, 8)$ , we then have V [ϱ] =95/24.

Case 2: $\bar{m} \leq e < p$ . Through similar procedures like Case 1, we eventually get $\begin{matrix} V [ϱ] & = \frac{(p - 2 \bar{m} + e) (i + \underline{m} - 3 \bar{m} + p)}{8 (p - \bar{m})} \\ + \frac{(i + \underline{m} - 3 \bar{m} + p)^{3}}{192 (p - \bar{m})} + \frac{(i - 3 \underline{m} + \bar{m} + p)^{2}}{32} \\ - \frac{(i + \underline{m} - 3 \bar{m} + p)^{2}}{32} + \frac{(i - 3 \underline{m} + \bar{m} + p)^{3}}{192 (\underline{m} - i)} \\ - \frac{(e - i) (i - 3 \underline{m} + \bar{m} + p)^{2}}{32 (\underline{m} - i)} + \frac{(e - i)^{3}}{6 (\underline{m} - i)} . \end{matrix}$ (15)

Provided that $ϱ \sim T (1, 2, 3, 10)$ , we then have V [ϱ] =223/42.

Example 11. Let $ϱ \sim (i, \underline{m}, \bar{m})$ be a Gaussian fuzzy interval with finite expected value e, where $e = (\underline{m} + \bar{m}) / 2$ from Example 8. In order to get the variance of ϱ, similarly, denote υ = (ϱ - e) ², and the membership function of υ can be represented by $μ_{υ} (x) = {\begin{matrix} 1, & if 0 \leq x < (\bar{m} - e)^{2} \\ e^{- (\frac{\sqrt{x} - \bar{m} + e}{i})^{2}}, & if x \geq (\bar{m} - e)^{2} \\ 0, & otherwise, \end{matrix}$ (16) as shown in Figure 9.

Fig.9

The membership function of υ in Example 9.

Then, based on Equations (2) and (24), we have

$\begin{matrix} {υ \geq t} = 1 - {υ < t} = \\ {\begin{matrix} \frac{1}{2}, & if 0 \leq t < (\bar{m} - e)^{2} \\ \frac{1}{2} e^{- (\frac{\sqrt{t} - \bar{m} + e}{i})^{2}}, & if t \geq (\bar{m} - e)^{2} . \end{matrix} \end{matrix}$ (17)

Finally, we can get the variance of ϱ via Equations (17) and (25), that is,

$\begin{matrix} V [ϱ] & = \int_{0}^{+ \infty} (1 - {υ < t}) d t \\ = \int_{0}^{(\bar{m} - e)^{2}} \frac{1}{2} d t + \int_{(\bar{m} - e)^{2}}^{+ \infty} \frac{1}{2} e^{- (\frac{\sqrt{t} - \bar{m} + e}{i})^{2}} d t \\ = \frac{1}{2} (\bar{m} - e)^{2} + \frac{1}{2} (i^{2} + i (\bar{m} - e) \sqrt{π}) . \end{matrix}$ (18)

Provided that ϱ ∼ G (1, 3, 5), we have $V [ϱ] = 1 + \sqrt{π} / 2 .$

4. The variance bounds for fuzzy interval

With respect to the trapezoidal fuzzy number and the Gaussian fuzzy interval mentioned in Sections 2 and 3, two relatively simple types among all fuzzy intervals, it seems that the calculation formulas for variance presented in Equations (11), (14), (15), (18) are not very simple. Considering that in many optimization problems it is often required to minimize the variance of functions involving a set of fuzzy numbers, and in such situation, it would be quite hard to get the exact variance especially when the functions are nonlinear, Yi et al. [19] suggested an upper bound as a substitute for fuzzy variance as follows.

Definition 4. (Yi et al. [19]) Let ϱ be a fuzzy number with expected value e and credibility distribution Γ. The upper bound of the variance (UBV) is defined by $\bar{V} [ϱ] = \int_{0}^{+ \infty} (1 - Γ (e + \sqrt{x}) + Γ (e - \sqrt{x})) x .$

Based on Definition 4, by using the inverse credibility distribution, two simple calculation formulas of UBV were put forward by Yi et al. [19], by using which the UBV for fuzzy numbers with continuous and strictly monotonous inverse credibility distributions (the so-called regular fuzzy numbers) can be effortlessly obtained and then utilized to substitute the exact variance in practical problems. Motivated by the work in [19], in this section, we first propose a simple calculation formula for UBV of regular LR fuzzy intervals. Note that the regular fuzzy number discussed in [19] is only a special case of regular LR fuzzy interval treated in this paper.

Theorem 5. Let ϱ be a regular LR fuzzy interval with expected value e and credibility distribution Γ. Then its UBV can be computed by

\bar{V} [ϱ] = \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β,

where Γ^-1 is the inverse credibility distribution of ϱ. It follows from Definition 4 that $\begin{matrix} \bar{V} [ϱ] & = \int_{0}^{+ \infty} (1 - Γ (e + \sqrt{x}) + Γ (e - \sqrt{x})) x \\ = \int_{0}^{+ \infty} d x - \int_{0}^{+ \infty} Γ (e + \sqrt{x}) d x \\ + \int_{0}^{+ \infty} Γ (e - \sqrt{x}) d x . \end{matrix}$ (19) Since the value of e may be located in the closed interval $[\underline{m}, \bar{m}]$ or not, three cases are discussed separately in the following.Case 1: $e < \underline{m}$ . In this case, by replacing $Γ (e + \sqrt{x})$ with β, $\int_{0}^{+ \infty} Γ (e + \sqrt{x}) d x$ in Eq. (29) can be reformed as $\begin{matrix} \int_{0}^{+ \infty} Γ (e + \sqrt{x}) d x \\ = \int_{0}^{(\underline{m} - e)^{2}} Γ (e + \sqrt{x}) d x + \int_{(\underline{m} - e)^{2}}^{(\bar{m} - e)^{2}} Γ (e + \sqrt{x}) d x \\ + \int_{(\bar{m} - e)^{2}}^{+ \infty} Γ (e + \sqrt{x}) d x \\ = \int_{Γ (e)}^{0.5} β d (Γ^{- 1} (β) - e)^{2} + \int_{0.5}^{1} β d (Γ^{- 1} (β) - e)^{2} \\ + \frac{1}{2} ((\bar{m} - e)^{2} - (\underline{m} - e)^{2}) . \end{matrix}$ Then, based on the integration by parts, we have $\begin{matrix} \int_{0}^{+ \infty} & Γ (e + \sqrt{x}) d x \\ = \frac{1}{2} (\bar{m} - e)^{2} + β (Γ^{- 1} (β) - e)^{2} |_{0.5}^{1} \\ - \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β . \end{matrix}$ (20) Similarly, by replacing $Γ (e - \sqrt{x})$ with β, $\int_{0}^{+ \infty} Γ (e - \sqrt{x}) d x$ in Eq. (29) can be changed into $\int_{0}^{+ \infty} Γ (e - \sqrt{x}) d x = \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β .$ (21) Substituting Equations (21) and (22) into Eq. (29), then we have $\begin{matrix} \bar{V} [ϱ] \\ = x |_{0}^{+ \infty} - \frac{1}{2} (\bar{m} - e)^{2} + \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ - β (Γ^{- 1} (β) - e)^{2} |_{0.5}^{1} + \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β \\ = - \frac{1}{2} (\bar{m} - e)^{2} + \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ + \frac{1}{2} (\bar{m} - e)^{2} . \end{matrix}$ (22) Simplifying Eq. (23), we finally have $\bar{V} [ϱ] = \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β .$ (23) As to the other two cases that $\underline{m} \leq e \leq \bar{m}$ and $e > \bar{m}$ , Eq. (5) can be derived through similar procedures like Case 1.□

The UBV advised in [19] can be reasonably applied to the situations that aim at minimizing the fuzzy variance, for example risk management of investors with risk-averse attitudes. And from another point of view, with respect to risk seekers, it is naturally needed to maximize the deviation in order for possible high returns. In allusion to this situation, we propose a new concept of lower bound of the variance on the basis of the following theorem.

Theorem 6. Let ϱ be a fuzzy number with finite expected value e and credibility distribution Γ. Then we have $V [ϱ] \geq \frac{1}{2} \int_{0}^{+ \infty} (1 - Γ (e + \sqrt{x}) + Γ (e - \sqrt{x})) d x .$ It follows from the definition of variance (see Eq. (17)) that $V [ϱ] = \int_{0}^{+ \infty} {(ϱ - e)^{2} \geq x} d x .$ In view of the self-duality of Cr, V [ϱ] can be then reformed as $\begin{matrix} V [ϱ] & = \int_{0}^{+ \infty} (1 - {(ϱ - e)^{2} < x}) d x \\ = \int_{0}^{+ \infty} (1 - {e - \sqrt{x} < ϱ < e + \sqrt{x}}) d x . \end{matrix}$ (24) Afterwards, since Cr is sub-additive, it follows that ${e - \sqrt{x} < ϱ < e + \sqrt{x}} \leq {ϱ < e + \sqrt{x}},$ and ${e - \sqrt{x} < ϱ < e + \sqrt{x}} \leq {ϱ > e - \sqrt{x}},$ respectively. That is, $\begin{matrix} {e - \sqrt{x} < ϱ < e + \sqrt{x}} \\ \leq \frac{1}{2} ({ϱ < e + \sqrt{x}} + {ϱ > e - \sqrt{x}}) . \end{matrix}$ (25) Substituting Eq. (6) into Eq. (25), we finally have $\begin{matrix} V [ϱ] & = \int_{0}^{+ \infty} (1 - {e - \sqrt{x} < ϱ < e + \sqrt{x}}) d x \\ \geq \int_{0}^{+ \infty} (1 - \frac{1}{2} ({ϱ < e + \sqrt{x}} \\ + {ϱ > e - \sqrt{x}})) d x \\ = \frac{1}{2} \int_{0}^{+ \infty} (1 - Γ (e + \sqrt{x}) + Γ (e - \sqrt{x})) d x . \end{matrix}$ □

Definition 5. Let ϱ be a regular LR fuzzy interval with finite expected value e and credibility distribution Γ. Then the lower bound of the variance (LBV) is defined by $\underline{V} [ϱ] = \frac{1}{2} \int_{0}^{+ \infty} (1 - Γ (e + \sqrt{x}) + Γ (e - \sqrt{x})) d x .$

From Definitions 4 and 5, it is easy to derive that $\underline{V} [ϱ] = \frac{1}{2} \bar{V} [ϱ]$ .

Theorem 7. Let ϱ be a regular LR fuzzy interval with credibility distribution Γ. If its expected value e exists, then the LBV of ϱ can be computed by

\underline{V} [ϱ] = \frac{1}{2} \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β,

where Γ^-1 is the inverse credibility distribution of ϱ.

It follows immediately from Theorem 5 and the relationship $\underline{V} [ϱ] = \frac{1}{2} \bar{V} [ϱ]$ .□

In accordance with Theorems 5 and 7, the UBV and LBV of regular LR fuzzy intervals can be effortlessly deduced by taking the trapezoidal fuzzy number and Gaussian fuzzy interval as examples.

Example 12. Let $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ be a trapezoidal fuzzy number with expected value $e = (i + \underline{m} + \bar{m} + p) / 4$ . Then its UBV and LBV can be deduced immediately through Theorems 5 and 7 as

$\begin{matrix} \bar{V} [ϱ] = \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ = & \int_{0}^{0.5} (2 (\underline{m} - i) β + i - e)^{2} d β \\ + \int_{0.5}^{1} (2 (p - \bar{m}) β + 2 \bar{m} - p - e)^{2} d β \\ = & \frac{7}{6} (p - \bar{m})^{2} + \frac{1}{2} (2 \bar{m} - p - e) (2 p - \bar{m} - e) \\ + \frac{(\underline{m} - i)^{2}}{6} + \frac{1}{2} (i - e) (\underline{m} - e) \end{matrix}$ (26) and $\underline{V} [ϱ] = \frac{1}{2} \bar{V} [ϱ]$ , respectively. Especially, when ϱ is a symmetric trapezoidal fuzzy number (i.e., $\underline{m} - i = p - \bar{m}$ ), $\bar{V} [ϱ]$ in Eq. (39) can be rewritten as $\begin{matrix} \bar{V} [ϱ] = & \frac{4}{3} (p - \bar{m})^{2} + \frac{1}{2} (2 \bar{m} - p - e) (2 p - \bar{m} - e) \\ + \frac{1}{2} (i - e) (\underline{m} - e) . \end{matrix}$

Provided that $ϱ \sim T (2, 4, 8, 10)$ , we have $\bar{V} [ϱ] = 28 / 3$ and $\underline{V} [ϱ] = 14 / 3 .$

Example 13. Let $ϱ \sim (i, \underline{m}, \bar{m})$ be a Gaussian fuzzy interval with expected value $e = (\underline{m} + \bar{m}) / 2$ . Then its UBV and LBV can be represented as $\begin{matrix} \bar{V} [ϱ] & = \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ = \int_{0}^{0.5} (\underline{m} - i \sqrt{- ln 2 β} - e)^{2} d β \\ + \int_{0.5}^{1} (\bar{m} + i \sqrt{- ln (2 - 2 β)} - e)^{2} d β \\ = i^{2} + (\underline{m} - e)^{2} + \frac{\sqrt{π}}{2} i (\bar{m} - \underline{m}) \end{matrix}$ and $\underline{V} [ϱ] = \frac{1}{2} \bar{V} [ϱ]$ , respectively. Provided that ϱ ∼ G (1, 3, 5), we then have $\bar{V} [ϱ] = 2 + \sqrt{π}$ and $\underline{V} [ϱ] = 1 + \sqrt{π} / 2$ .

As mentioned above, it is oftentimes required to minimize or maximize the variance of functions, V [f (ξ₁, ξ₂, ⋯ , ξ_n)], involving a set of fuzzy intervals ξ₁, ξ₂, ⋯ , ξ_n during the decision making of optimization problems. Owing to the computational complexity of fuzzy variance according to Eq. (17), we suggest turning to minimize $\bar{V} [f (ξ_{1}, ξ_{2}, \dots, ξ_{n})]$ or maximize $\underline{V} [f (ξ_{1}, ξ_{2}, \dots, ξ_{n})]$ alternatively. In order to deal with these situations, the following calculation formulas are thus raised.

Theorem 8. Let ϱ₁, ϱ₂, ⋯ , ϱ_n be independent regular LR fuzzy intervals with credibility distributions Γ₁, Γ₂, ⋯ , Γ_n, respectively. If the function f (x₁, x₂, ⋯ , x_n) is continuous and strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the UBV and LBV of ϱ = f (ϱ₁, ϱ₂, ⋯ , ϱ_n) are

\begin{matrix} \bar{V} [ϱ] & = \int_{0}^{1} (f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) - e)^{2} d β, \\ \underline{V} [ϱ] & = \frac{1}{2} \int_{0}^{1} (f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) - e)^{2} d β, \end{matrix}

respectively, where e is the expected value of ϱ calculated by Eq. (15) in Theorem 3, and $Γ_{1}^{- 1}, Γ_{2}^{- 1}, \dots, Γ_{n}^{- 1}$ are the inverse credibility distributions of ϱ₁, ϱ₂, ⋯ , ϱ_n, respectively.

Since {ϱ₁, ϱ₂, ⋯ , ϱ_n} is a set of independent regular LR fuzzy intervals and the function f is continuous and strictly monotonous, it follows from Theorem 1 that the inverse credibility distribution of ϱ can be represented as $\begin{matrix} Ψ_{ϱ}^{- 1} (β) = & f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) . \end{matrix}$ After that, the UBV and LBV of ϱ are then derived according to Theorems 5 and 7 straightforwardly.□

To sum up, in this section, we first derive a calculation formula of the upper bound of the variance defined in [19] for regular LR fuzzy intervals. Subsequently, a lower bound of the variance is initiated for fuzzy variance from the perspective of optimization, and simultaneously a simple calculation formula is put forward. Furthermore, in order to tackle the decision making problems like min V [f (ξ₁, ξ₂, ⋯ , ξ_n)] or max V [f (ξ₁, ξ₂, ⋯ , ξ_n)] in real applications, Theorem 8 is advised.

5. Semi-variances for fuzzy intervals

In the original definition of variance in [12] (see Eq. (16)), it is obvious that the positive and negative deviations are treated equally, which is not always suitable for all situations. For instance, in stock portfolio selection, most investors have completely different attitudes to positive deviation and negative deviation. From the angle of loss aversion, the negative deviation is expected to be minimized. Starting from this idea, Huang [8] proposed a conception of downside semi-variance as follows.

Definition 6. (Huang [8]) Let ϱ be a fuzzy number with finite expected value e. Then the downside semi-variance of ϱ is defined by ${Sv}^{-} [ϱ] = E [((ϱ - e)^{-})^{2}],$ where $(ϱ - e)^{-} = {\begin{matrix} ϱ - e, & if ϱ \leq e \\ 0, & if ϱ > e . \end{matrix}$

Based upon the definition, two fuzzy mean-semi-variance models were designed by Huang [8], which have been applied to portfolio selection problems (see, e.g., [17]). Since the calculation processes for downside semi-variance are quite complex as shown in [8], a simpler calculation formula is then suggested specifically for regular LR fuzzy intervals as follows.

Theorem 9. Let ϱ be a regular LR fuzzy interval with credibility distribution Γ. If its expected value e exists, then the downside semi-variance can be calculated by

{Sv}^{-} [ϱ] = \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β,

(27)

where Γ^-1 is the inverse credibility distribution of ϱ.

Denote the lower and upper interval limits of ϱ as $\underline{m}$ and $\bar{m}$ , respectively. It follows from Definitions 3 and 6 that $\begin{matrix} {Sv}^{-} [ϱ] & = \int_{0}^{+ \infty} {(ϱ - e)^{2} \geq t, ϱ \leq e} d t \\ + \int_{0}^{+ \infty} {0 \geq t, ϱ > e} d t \\ = \int_{0}^{+ \infty} {ϱ \leq e - \sqrt{t}} d t + 0 \\ = \int_{0}^{+ \infty} Γ (e - \sqrt{t}) t . \end{matrix}$ (28) Similarly, according to the relationship between the expected value e and the closed interval $[\underline{m}, \bar{m}]$ , three cases are discussed respectively in thefollowing. Case 1: $e < \underline{m}$ . In this case, a similar proof procedure for Theorem 5 can be utilized to prove the equality in Eq. (46). Case 2: $\underline{m} \leq e \leq \bar{m}$ . In this case, when replacing $Γ (e - \sqrt{t})$ with β, Eq. (47) can be transformed to $\begin{matrix} {Sv}^{-} [ϱ] \\ = \int_{0}^{(e - \underline{m})^{2}} Γ (e - \sqrt{t}) d t + \int_{(e - \underline{m})^{2}}^{+ \infty} Γ (e - \sqrt{t}) d t \\ = \frac{1}{2} (e - \underline{m})^{2} + β (Γ^{- 1} (β) - e)^{2} |_{Γ (\underline{m})}^{0} \\ - \int_{Γ (\underline{m})}^{0} (Γ^{- 1} (β) - e)^{2} d β . \end{matrix}$ By simplifying the expression, we have ${Sv}^{-} [ϱ] = \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β .$ Case 3: $e > \bar{m}$ . Similarly to the proof procedure of Case 2, we can deduce that $\begin{matrix} {Sv}^{-} [ϱ] \\ = & \int_{0}^{(e - \bar{m})^{2}} Γ (e - \sqrt{t}) d t + \int_{(e - \bar{m})^{2}}^{(e - \underline{m})^{2}} Γ (e - \sqrt{t}) d t \\ + \int_{(e - \underline{m})^{2}}^{+ \infty} Γ (e - \sqrt{t}) d t \\ = & \int_{Γ (\underline{m})}^{0} β d (Γ^{- 1} (β) - e)^{2} + \int_{Γ (e)}^{Γ (\bar{m})} β d (Γ^{- 1} (β) - e)^{2} \\ + \frac{1}{2} ((e - \underline{m})^{2} - (e - \bar{m})^{2}) \\ = & \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β . \end{matrix}$ □

Downside semi-variance naturally gauges the negative deviation of a fuzzy number from its expected value, which may legitimately assist the people in the choice of decision making with both higher returns and less loss. Oppositely, in practical optimization problems, the objective is maybe to minimize the possible cost or loss. Under these circumstances, the positive deviation is usually undesirable. In order to model this type of problems, we recommend a new concept of upside semi-variance as follows.

Definition 7. Let ϱ be a fuzzy number with finite expected value e. Then the upside semi-variance of ϱ is defined by ${Sv}^{+} [ϱ] = E [((ϱ - e)^{+})^{2}],$ where $(ϱ - e)^{+} = {\begin{matrix} ϱ - e, & if ϱ > e \\ 0, & if ϱ \leq e . \end{matrix}$

Theorem 10. Let ϱ be a regular LR fuzzy interval with credibility distribution Γ. If its expected valuee exists, then the upside semi-variance can be calculated by

{Sv}^{+} [ϱ] = \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β,

where Γ^-1 is the inverse credibility distribution of ϱ.

It follows immediately from Theorem 9.□

Theorem 11. Let ϱ be a regular LR fuzzy interval with finite expected value. If it is symmetric, then it’s upside and downside semi-variance satisfies ${Sv}^{-} [ϱ] = {Sv}^{+} [ϱ] .$ (29)

Denote the expected value of ϱ as e. It follows from Theorems 9 and 10 that $\begin{matrix} {Sv}^{+} & = \int_{0}^{+ \infty} {ϱ \leq e - \sqrt{t}} t, \\ {Sv}^{-} & = \int_{0}^{+ \infty} {ϱ \geq e + \sqrt{t}} t . \end{matrix}$ On the basis of Eq. (2), we have $\begin{matrix} {ϱ \leq e - \sqrt{t}} t & = \frac{1}{2} {ϱ \leq e - \sqrt{t}}, \\ {ϱ \geq e + \sqrt{t}} t & = \frac{1}{2} {ϱ \geq e + \sqrt{t}} . \end{matrix}$ Since ϱ is a symmetric regular LR fuzzy interval, it can be easily deduced that

{ϱ \leq e - \sqrt{t}} = {ϱ \geq e + \sqrt{t}} .

Eventually, we can obtain ${Sv}^{-} [ϱ] = {Sv}^{+} [ϱ] .$ □

Example 14. Let $ϱ \sim T (i, \underline{m}, \bar{m}, p)$ be a trapezoidal fuzzy number with expected value $e = (i + \underline{m} + \bar{m} + p) / 4$ . Then, from Theorems 9 and 10, the upside and downside semi-variances can be calculated by $\begin{matrix} {Sv}^{+} [ϱ] & = \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β, \\ {Sv}^{-} [ϱ] & = \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β . \end{matrix}$ That is, ${Sv}^{+} [ϱ] = {\begin{matrix} 2 (p - \bar{m}) (2 \bar{m} - p - e) (1 - (Γ (e))^{2}) + \\ \frac{4}{3} (p - \bar{m})^{3} (1 - (Γ (e))^{3}) + (2 \bar{m} - p - \\ e)^{2} (1 - Γ (e)), if e \geq \underline{m} \\ \frac{7}{6} (p - \bar{m})^{2} + \frac{1}{2} (2 \bar{m} - p - e)^{2} + \frac{3}{2} (p - \\ \bar{m}) (2 \bar{m} - p - e) + 2 (\underline{m} - i) (i - e) (\frac{1}{4} - \\ Γ (e)) + \frac{4}{3} (\underline{m} - i)^{2} (\frac{1}{8} - (Γ (e))^{3}) + \\ (i - e)^{2} (\frac{1}{2} - Γ (e)), if e < \underline{m}, \end{matrix}$ ${Sv}^{-} [ϱ] = {\begin{matrix} \frac{4}{3} (\underline{m} - i)^{3} (Γ (e))^{3} + (i - e)^{2} Γ (e) + \\ 2 (\underline{m} - i) (i - e) (Γ (e))^{2}, if e \leq \bar{m} \\ \frac{4}{3} (p - \bar{m})^{3} ((Γ (e))^{3} - \frac{1}{8}) + (2 \bar{m} - p - \\ e)^{2} (Γ (e) - \frac{1}{2}) + 2 (p - \bar{m}) (2 \bar{m} - p - \\ e) ((Γ (e))^{2} - \frac{1}{4}) + \frac{1}{6} (\underline{m} - i)^{2} + \\ \frac{1}{2} (i - e)^{2} + (\underline{m} - i) (i - e), if e > \bar{m}, \end{matrix}$ where Γ (e) is the credibility of ϱ ≤ e and can be computed by Eq. (6).

Provided that $ϱ \sim T (1, 2, 5, 8)$ , we then have Sv⁺ [ϱ] =7/2 and Sv^- [ϱ] =19/6.

Example 15. Let $ϱ \sim (i, \underline{m}, \bar{m})$ be a Gaussian fuzzy interval with expected value $e = (\underline{m} + \bar{m}) / 2$ . Then its upside and downside semi-variances are $\begin{matrix} {Sv}^{+} & [ϱ] = {Sv}^{-} [ϱ] \\ = \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ = \frac{1}{2} (\bar{m} - e)^{2} + \frac{i^{2}}{2} + \frac{\sqrt{π}}{2} i (\bar{m} - e) . \end{matrix}$ Provided that $ϱ \sim T (1, 3, 5)$ , we then have ${Sv}^{+} [ϱ] = {Sv}^{-} [ϱ] = 1 + \sqrt{π} / 2$ .

Theorem 12. Let ϱ₁, ϱ₂, ⋯ , ϱ_n be independent regular LR fuzzy intervals with credibility distributions Γ₁, Γ₂, ⋯ , Γ_n, respectively. If the function f (x₁, x₂, ⋯ , x_n) is continuous and strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the upside and downside semi-variances of ϱ = f (ϱ₁, ϱ₂, ⋯ , ϱ_n) are

\begin{matrix} {Sv}^{+} [ϱ] = & \int_{Γ (e)}^{1} (f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) - e)^{2} d β, \\ {Sv}^{-} [ϱ] = & \int_{0}^{Γ (e)} (f (Γ_{1}^{- 1} (β), \dots, Γ_{m}^{- 1} (β), \\ Γ_{m + 1}^{- 1} (1 - β), \dots, Γ_{n}^{- 1} (1 - β)) - e)^{2} d β, \end{matrix}

It follows immediately from Theorems 1,9-10.□

In summary, this section first derives a simple calculation formula of the downside semi-variance defined in [8] for regular LR fuzzy intervals, and then recommends a novel concept of upside semi-variance as well as a calculation formula in order for modeling the problems with undesirability of positive deviations. Additionally, in order to calculate the semi-variances of functions involving a set of independent regular LR fuzzy intervals, Theorem 12 is given in the light of the operational law of inverse credibility distribution and the calculation formulas presented above.

6. Inequalities

In this section, for the purpose of better illustrating V [ϱ], $\bar{V} [ϱ]$ , $\underline{V} [ϱ]$ , Sv⁺ [ϱ], and Sv^- [ϱ], the relationships among them are further discussed. After that, some related inequalities are presented and proved for efficiently handling the decision-making problems in practice.

6.1. Relationships among V [ϱ], $\bar{V} [ϱ]$ , $\underline{V} [ϱ]$ , Sv⁺ [ϱ], and Sv^- [ϱ]

Theorem 13. Let ϱ be a regular LR fuzzy interval with finite expected value. We then have

(a) $V [ϱ] < \bar{V} [ϱ]$ ;

(b) $V [ϱ] \geq \underline{V} [ϱ]$ ;

(d) Sv⁺ [ϱ] ≤ V [ϱ];

(e) ${Sv}^{+} [ϱ] + {Sv}^{-} [ϱ] = \bar{V} [ϱ] = 2 \underline{V} [ϱ] .$

Proof. For convenience, denote the expected value and the credibility distribution of ϱ as e and Γ, respectively.

(a) and (b) It follows immediately from Definitions 4 and 5, and Eq. (17).

(c) It follows from the definition of variance that $V [ϱ] = \int_{0}^{+ \infty} {{ϱ - e \geq \sqrt{t}} \cup {ϱ - e \leq - \sqrt{t}}} d t .$ Afterwards, on the basis of Theorem 9, we have $\begin{matrix} {Sv}^{-} [ϱ] = \int_{0}^{+ \infty} {ϱ \leq e - \sqrt{t}} d t . \end{matrix}$ For any t ≥ 0, since Cr is subadditive, then we have $\begin{matrix} Cr {{ϱ - e \geq \sqrt{t}} \cup {ϱ - e \leq - \sqrt{t}}} \\ \geq Cr {ϱ \leq e - \sqrt{t}}, \end{matrix}$ which implies that Sv^- [ϱ] ≤ V [ϱ] .

(d) A similar proof procedure to part (b) can derive Sv⁺ [ϱ] ≤ V [ϱ] from Theorem 10, Eq. (17) and the subadditivity of Cr.

(e) Following from Theorems 5, 7, 9, and 10, we have $\begin{matrix} {Sv}^{-} [ϱ] & = \int_{0}^{Γ (e)} (Γ^{- 1} (β) - e)^{2} d β, \\ {Sv}^{+} [ϱ] & = \int_{Γ (e)}^{1} (Γ^{- 1} (β) - e)^{2} d β, \\ \underline{V} [ϱ] & = \frac{1}{2} \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β, \\ \bar{V} [ϱ] & = \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β, \end{matrix}$ which follows immediately that $\begin{matrix} {Sv}^{+} [ϱ] + {Sv}^{-} [ϱ] = & \int_{0}^{1} (Γ^{- 1} (β) - e)^{2} d β \\ = & \bar{V} [ϱ] = 2 \underline{V} [ϱ] . \end{matrix}$ □

6.2. Some inequalities for fuzzy intervals

Theorem 14. Let ϱ and υ be two independent regular LR fuzzy intervals. Then we have (a) $\bar{V} [ϱ + υ] \leq 2 (\bar{V} [ϱ] + \bar{V} [υ])$ , (b) $\underline{V} [ϱ + υ] \leq 2 (\underline{V} [ϱ] + \underline{V} [υ])$ , and the equalities hold if and only if Γ (x) = Ψ (x + k), where k is a constant.

Proof. (a) Denote f (x₁, x₂) = x₁ + x₂, and the credibility distributions of ϱ and υ by Γ and Ψ, respectively. Then from Theorem 8, we have $\begin{matrix} \bar{V} & [ϱ + υ] \\ = \int_{0}^{1} (f (Γ^{- 1} (β), Ψ^{- 1} (β)) - E [ϱ + υ]) d r \\ = \int_{0}^{1} ((Γ^{- 1} (β) + Ψ^{- 1} (β)) - E [ϱ + υ])^{2} d β . \end{matrix}$ Besides, with respect to E [ϱ + υ], taking Theorem 4 for reference, we have $E [ϱ + υ] = E [ϱ] + E [υ] .$ Then the expression $\bar{V} [ϱ + υ]$ can be reformed as $\begin{matrix} \bar{V} [ϱ + υ] \\ = & \int_{0}^{1} ((Γ^{- 1} (β) + Ψ^{- 1} (β)) - E [ϱ + υ])^{2} d β \\ = & \int_{0}^{1} ((Γ^{- 1} (β) - E [ϱ]) + (Ψ^{- 1} (β) - E [υ]))^{2} d β \\ \leq & \int_{0}^{1} 2 ((Γ^{- 1} (β) - E [ϱ])^{2} + (Ψ^{- 1} (β) - E [υ])^{2}) d β \\ = & 2 (\bar{V} [ϱ] + \bar{V} [υ]) . \end{matrix}$ Note that the equality holds if and only if Γ^-1 (β) - E [ϱ] = Ψ^-1 (β) - E [υ]. Denoting that k = E [υ] - E [ϱ], then we have Γ (x) = Ψ (x + k).(b) It follows immediately from part (a) and $\underline{V} [ϱ] = \frac{1}{2} \bar{V} [ϱ]$ .□

Example 16. Let $ϱ \sim (i_{1}, {\underline{m}}_{1}, {\bar{m}}_{1})$ and $υ \sim (i_{2}, {\underline{m}}_{2}, {\bar{m}}_{2})$ be two independent Gaussian fuzzy intervals with expected values $E [ϱ] = ({\underline{m}}_{1} + {\bar{m}}_{1}) / 2$ and $E [υ] = ({\underline{m}}_{2} + {\bar{m}}_{2}) / 2$ , respectively. Then from Theorem 1, the inverse credibility distribution of ϱ + υ, denoted as Γ^-1 (β), can be written as $\begin{matrix} Γ^{- 1} (β) = Γ_{1}^{- 1} (β) + Γ_{2}^{- 1} (β) = \\ {\begin{matrix} {\underline{m}}_{1} + {\underline{m}}_{2} - (i_{1} + i_{2}) \sqrt{- ln (2 β)}, \\ if β < 0.5 \\ [{\underline{m}}_{1} + {\underline{m}}_{2}, {\bar{m}}_{1} + {\bar{m}}_{2}], \\ if β = 0.5 \\ {\bar{m}}_{1} + {\bar{m}}_{2} + (i_{1} + i_{2}) \sqrt{- ln (2 - 2 β)}, \\ if β > 0.5, \end{matrix} \end{matrix}$ where $Γ_{1}^{- 1} (β)$ and $Γ_{2}^{- 1} (β)$ are the inverse credibility distributions of ϱ and υ, respectively. Based on Theorem 6, we can obtain the lower bound of V [ϱ + υ], that is, $\begin{matrix} \underline{V} [ϱ + υ] \\ = \int_{0}^{0.5} (({\underline{m}}_{1} + {\underline{m}}_{2}) - (i_{1} + i_{2}) \sqrt{- ln (2 β)} - e)^{2} d x \\ + \int_{0.5}^{1} (({\bar{m}}_{1} + {\bar{m}}_{2}) + (i_{1} + i_{2}) \sqrt{- ln (2 - 2 β)} - e)^{2} d x \\ = \frac{(i_{1} + i_{2})^{2}}{2} + \frac{({\underline{m}}_{1} + {\underline{m}}_{2} - e)^{2}}{2} + \frac{\sqrt{π}}{4} (i_{1} + i_{2}) ({\bar{m}}_{1} \\ + {\bar{m}}_{2} - {\underline{m}}_{1} - {\underline{m}}_{2}), \end{matrix}$ where e is the expected value of ϱ + υ with $e = E [ϱ] + E [υ] = \frac{{\underline{m}}_{1} + {\underline{m}}_{2} + {\bar{m}}_{1} + {\bar{m}}_{2}}{2} .$

Note that $\begin{matrix} \underline{V} [ϱ] & = \frac{i_{1}^{2}}{2} + \frac{({\underline{m}}_{1} - {\bar{m}}_{1})^{2}}{8} + \frac{\sqrt{π}}{4} i_{1} ({\bar{m}}_{1} - {\underline{m}}_{1}), \\ \underline{V} [υ] & = \frac{i_{2}^{2}}{2} + \frac{({\underline{m}}_{2} - {\bar{m}}_{2})^{2}}{8} + \frac{\sqrt{π}}{4} i_{2} ({\bar{m}}_{2} - {\underline{m}}_{2}) . \end{matrix}$

Then we have $\underline{V} [ϱ + υ] \leq 2 (\underline{V} [ϱ] + \underline{V} [υ]),$ and the equality holds if and only if ${\underline{m}}_{1} - {\bar{m}}_{1} = {\underline{m}}_{2} - {\bar{m}}_{2}$ and i₁ = i₂, which means $Γ_{1} (x) = Γ_{2} (x + ({\underline{m}}_{2} + {\bar{m}}_{2}) / 2 - ({\underline{m}}_{1} + {\bar{m}}_{1}) / 2)$ . Besides, in a similar way, we can also obtain $\bar{V} [ϱ + υ] \leq 2 (\bar{V} [ϱ] + \bar{V} [υ])$ .

Theorem 15. Let ϱ and υ be two independent regular LR fuzzy intervals. Then we have (a) $\sqrt{\bar{V} [ϱ + υ]} \leq \sqrt{\bar{V} [ϱ]} + \sqrt{\bar{V} [υ]}$ ; (b) $\sqrt{\underline{V} [ϱ + υ]} \leq \sqrt{\underline{V} [ϱ]} + \sqrt{\underline{V} [υ]}$ , and the equalities hold if and only if Γ (x) = Ψ (mx + n), where m and n are constants.

For convenience, denote the credibility distributions of ϱ and υ by Γ and Ψ, respectively (a) First, squaring both sides of the inequality, we get $\bar{V} [ϱ + υ] \leq \bar{V} [ϱ] + \bar{V} [υ] + 2 \sqrt{\bar{V} [ϱ]} \sqrt{\bar{V} [υ]} .$ From Theorem 5, the inequality can be reformed as $\begin{matrix} \int_{0}^{1} & ((Γ^{- 1} (β) - E [ϱ]) + (Ψ^{- 1} (β) - E [υ]))^{2} d β \\ \leq & \bar{V} [ϱ] + \bar{V} [υ] + 2 \sqrt{\bar{V} [ϱ]} \sqrt{\bar{V} [υ]} . \end{matrix}$ Simplifying this inequality, we then have $\begin{matrix} \int_{0}^{1} & (Γ^{- 1} (β) - E [ϱ]) (Ψ^{- 1} (β) - E [υ]) d β \\ \leq & \sqrt{\bar{V} [ϱ]} \sqrt{\bar{V} [υ]} . \end{matrix}$ Therefore, in order to prove part (a), it is necessary to prove $\int_{0}^{1} \frac{(Γ^{- 1} (β) - E [ϱ])}{\sqrt{\bar{V} [ϱ]}} \cdot \frac{(Ψ^{- 1} (β) - E [υ])}{\sqrt{\bar{V} [υ]}} d β \leq 1 .$ By considering the left side of the above inequality, it is easy to derive that $\begin{matrix} \int_{0}^{1} \frac{(Γ^{- 1} (β) - E [ϱ])}{\sqrt{\bar{V} [ϱ]}} \cdot \frac{(Ψ^{- 1} (β) - E [υ])}{\sqrt{\bar{V} [υ]}} d β \\ \leq & \frac{1}{2} \int_{0}^{1} \frac{(Γ^{- 1} (β) - E [ϱ])^{2}}{\bar{V} [ϱ]} + \frac{(Ψ^{- 1} (β) - E [υ])^{2}}{\bar{V} [υ]} d β \\ = & 1 . \end{matrix}$ Thus, the inequality in part (a) is proved. Note that the equality holds if and only if $\frac{Γ^{- 1} (β) - E [ϱ]}{\sqrt{\bar{V} [ϱ]}} = \frac{Ψ^{- 1} (β) - E [υ]}{\sqrt{\bar{V} [υ]}} .$ That is, $Ψ^{- 1} (β) = Γ^{- 1} (β) \sqrt{\frac{\bar{V} [υ]}{\bar{V} [ϱ]}} + E [υ] - E [ϱ] \sqrt{\frac{\bar{V} [υ]}{\bar{V} [ϱ]}} .$ Denote $m = \sqrt{\bar{V} [υ] / \bar{V} [ϱ]}$ , and $n = E [υ] - E [ϱ] \sqrt{\bar{V} [υ] / \bar{V} [ϱ]}$ , respectively, and then we have Ψ^-1 (β) = mΓ^-1 (β) + n, that is Γ (x) = Ψ (mx + n).(b) From Theorems 5 and 7 and the proof process for part (a), the inequality in part (b) holds obviously.□

Example 17. Let $ϱ \sim (i_{1}, {\underline{m}}_{1}, {\bar{m}}_{1})$ and $υ \sim (i_{2}, {\underline{m}}_{2}, {\bar{m}}_{2})$ be two independent Gaussian fuzzy intervals, respectively. From Example 16, we can obtain $\begin{matrix} \underline{V} = \frac{(i_{1} + i_{2})^{2}}{2} + \frac{({\underline{m}}_{1} + {\underline{m}}_{2} - e)^{2}}{2} \\ + \frac{\sqrt{π}}{4} (i_{1} + i_{2}) ({\bar{m}}_{1} + {\bar{m}}_{2} - {\underline{m}}_{1} - {\underline{m}}_{2}), \\ \underline{V} [ϱ] = \frac{i_{1}^{2}}{2} + \frac{({\underline{m}}_{1} - {\bar{m}}_{1})^{2}}{8} + \frac{\sqrt{π}}{4} i_{1} ({\bar{m}}_{1} - {\underline{m}}_{1}), \\ \underline{V} [υ] = \frac{i_{2}^{2}}{2} + \frac{({\underline{m}}_{2} - {\bar{m}}_{2})^{2}}{8} + \frac{\sqrt{π}}{4} i_{2} ({\bar{m}}_{2} - {\underline{m}}_{2}) . \end{matrix}$

Using the numerical results above, we can conduct that $\sqrt{\underline{V} [ϱ + υ]} \leq \sqrt{\underline{V} [ϱ]} + \sqrt{\underline{V} [υ]} .$ The equality holds if and only if i₁ = i₂ and ${\underline{m}}_{1} - {\bar{m}}_{1} = {\underline{m}}_{2} - {\bar{m}}_{2}$ . In addition, the inequality $\sqrt{\underline{V} [ϱ + υ]} \leq \sqrt{\underline{V} [ϱ]} + \sqrt{\underline{V} [υ]}$ can be also testified with a similar procedure.

7. Conclusion

The main contribution of this paper includes two aspects. Firstly, we extended the existing concepts related to fuzzy variance and simultaneously derived some relatively simple calculation formulas in order for effective and efficient utilization of variance in the practical applications. The main definitions and theorems presented in our paper were summarized in Table 1.

Table 1

Main definitions and theorems presented in our paper

Name	Notation	Definition	Calculation formula for fuzzy interval ϱ	Calculation formula for function of fuzzy intervals
Variance	V	Liu [12]	None	None
Upper bound of the variance	$\bar{V}$	Yi et al. [19]	Theorem 15	Theorem 8
Lower bound of the variance	$\underline{V}$	Definition 5	Theorem 7	Theorem 8
Upside semi-variance	Sv ⁺	Definition	Theorem 10	Theorem 12
Downside semi-variance	Sv ^-	Huang [8]	Theorem 9	Theorem 12

Secondly, some relevant inequalities and equalities of V, $\bar{V}$ , $\underline{V}$ , Sv⁺, and Sv^- are further discussed based upon their definitions and calculation formulas as well, including their relationship and some related inequalities. Besides, three typical and commonly-used LR fuzzy intervals, i.e., the symmetric and asymmetric trapezoidal fuzzy numbers, the Gaussian fuzzy interval, are employed throughout this paper for distinctly illustrating the definitions, calculation formulas, and correlation inequalities.

Finally, it should be noted that the derived calculation formulas in this paper, together with the deduced inequalities, for V, $\bar{V}$ , $\underline{V}$ , Sv⁺, and Sv^- can be only used for continuous and strictly monotone functions of regular LR fuzzy intervals. In order to widen their applicability, future work will focus on the calculation of the four variances in more complex situations, for example, general fuzzy intervals, more complex functions, and so on. In addition, it is also attractive to apply this work to model fuzzy portfolio selection problems, optimization problems concerning minimum possible cost or loss, and risk management problems, which may significantly decrease the calculation complexity and return satisfactory results.

Footnotes

Acknowledgments

The authors are grateful to the associate editor and the anonymous referees for their valuable comments and suggestions that significantly improved the paper.

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Calculation formulas and correlation inequalities for variance bounds and semi-variances of fuzzy intervals

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Fuzzy interval

6.1. Relationships among V [ϱ], V ¯ [ ϱ ] , V _ [ ϱ ] , Sv+ [ϱ], and Sv- [ϱ]

6.2. Some inequalities for fuzzy intervals

7. Conclusion

Footnotes

Acknowledgments

References

6.1. Relationships among V [ϱ], $\bar{V} [ϱ]$ , $\underline{V} [ϱ]$ , Sv⁺ [ϱ], and Sv^- [ϱ]