1 Introduction

Uncertainty occurs due to lack of precision, deficiency in data, diminutive sample sizes, foreseeable man-made/artificial mistakes etc., is an unavoidable component of real world problems. To deal with this type of uncertainty fuzzy set theory (FST) [1] is explored. In power system planning reliability investigation is an extremely significant feature. The electrical energy production and consumption are the essential operating characteristic of the power system those are operated simultaneously and consequently, the investigation of reliability obligation for power system is incredibly elevated. Generally probabilistic approaches of reliability investigation are explored. However, due to association of uncertainty in the system classical probability approaches are seemed to be inappropriate and subsequently, fuzzy reliability investigation models are taken into consideration [2]. Some recent applications in reliability investigations can be found in [3,4,5,6,7,8,9,10,11,12,13,14,15]. On the other hand, in medical diagnosis, usually a disease is characterized by several unswervingly perceptible symptoms which persuade the patient to visit a consultant or practitioner. A set of clinical inspections are commenced to make out the incidence of a disease. In the sphere of medical diagnosis, plenty of variables persuade the decision making process and subsequently, discriminate the judgments of the consultant or practitioner. Furthermore, mainly the medical diagnosis quandary engages dealing with uncertainties and so needed to integrate all the information into investigation. Therefore, fuzzy sets are explored to represent uncertainty and to perform medical diagnosis as well [16]. Some recent development in medical diagnosis can also be encountered in [17,18,19,20,21,22,23,24]. Afterwards Zadeh [25] himself developed possibility theory which was thought to be better to treat uncertainty and further studied by acolyte researchers such as Dubois and Prade [26], Klir [27], Yager [28] etc. Furthermore, Dubois and Prade [29] studied mean value of fuzzy numbers, Ban [30] discussed fuzzy valued measure and conditional expectations of fuzzy numbers, Heilpern [31]studied expected value of fuzzy numbers, Carlsson and Fuller [32] developed possiblistic mean and variance of fuzzy numbers, Chen and Tan [33] further developed mean value and variance of multiplication of fuzzy numbers.

Nevertheless in the absence of self duality measure the earlier studies lead to the exaggeration of the reality. keeping this in mind, Liu and Liu [34] initiated a concept termed as credibility theory. Li and Liu [35] presented a sufficient and necessary condition for credibility measures. Further Liu and Liu [36] systematically studied and developed credibility theory. After that some extended studies on credibility theory can be observed in Liu [37], Zhou et al. [38], Yi et al. [39], Garai et al. [40].

Although various types of fuzzy variables are encountered, however, an exceptional and intricate fuzzy variable SEFV in terms of credibility theory is not deliberated yet. This paper presents an approach to derive possibility, necessity and credibility measure of the SEFV. Furthermore, expected value, variance, rational upper bound etc of SEFV are presented. Then ranking of SEFNs through expected value and variance is proposed. Finally, novelty and applicability has been exhibited by performing reliability analysis and medical diagnosis cases.

2 Preliminaries

Uncertainty is an important as well as unavoidable ingredient of decision making process. Depending on the nature and accessibility of data, information, uncertainty is generally modelled using fuzzy set, possibility theory and Credibility theory.

Definition

Let \(\Theta\) be a non-empty set, and P the power set of \(\Theta\), and Pos is possibility measure. Then, the triplet \((\Theta , P, Pos)\) is known as a possibility space. A fuzzy variable is a mapping from possibility space \((\Theta , P, Pos)\) to the set of real numbers [37, 39].

Definition

Let \(\zeta\) be a fuzzy variable defined on the credibility space \((\Theta , P, Pos)\). Then its membership function (MF) defined from the credibility measure is given by [37, 39]

$$\begin{aligned} \mu _\zeta (x)=min(2Cr{\zeta =1});x\in {\mathbb {R}}. \end{aligned}$$

Definition

The \(\alpha\)-cut of a fuzzy variable A is defined as

$$\begin{aligned} ^{\alpha }A=\left\{ x \in X:\mu _A(x)\ge \alpha \right\} . \end{aligned}$$

Definition

Let A be a fuzzy variable, \(\mu\) be the MF of A, and r be any real number. Then, the possibility measure of A is defined as [25]

$$\begin{aligned} Pos\left\{ A\le r\right\} =\underset{x\le r}{sup}\,\mu _{A}(x) \end{aligned}$$

Definition

Let A be a fuzzy variable, \(\mu\) be the MF of A, and r be any real number. Then, the possibility measure of A is defined as [25]

$$\begin{aligned} Nec\left\{ A\le r\right\} =1-\underset{x>r}{sup}\,\mu _{A}(x) \end{aligned}$$

Definition

A credibility measure (Cr) is a non–negative set function holds the following [35]

  • \(Cr(\Theta )=1\)

  • \(Cr(A)\le Cr(B)\) for whenever \(A\subset B\)

  • \(Cr(A) + Cr(A^c)=1\) for any A

  • \(Cr\left\{ \cup A\right\} =\underset{i}{Sup}Cr \{A_i\}\) for any events \({A_i}\) with \(\underset{i}{Sup}Cr\{A_i\}<0.5\)

If the fuzzy variable A is given by its MF \(\mu\), then

$$\begin{aligned} Cr(A\le r)=\dfrac{1}{2}\left\{ Sup_{x\le r}\mu _A (x) +1 - Sup_{x> r}\mu _A (x) \right\} ,x,y\in {\mathbb {R}} \end{aligned}$$

Definition

The credibility distribution \(\Phi _A:{\mathbb {R}}\rightarrow [0,1]\) of a fuzzy variable A is defined as [36]

$$\begin{aligned} \Phi _A(x)=Cr \{\theta \in \Theta :\zeta (\theta )\le x\}. \end{aligned}$$

That is, the credibility that the fuzzy variable \(\zeta\) takes a value less than or equal to x.

Definition

The credibility density function of credibility distribution defined as \(\phi _A:{\mathbb {R}} \rightarrow [o,\infty )\) of any fuzzy variable A, is a function such that [37]

$$\begin{aligned} \phi _A (x)= \int _{\infty }^x \phi (y)dy, \forall x\in {\mathbb {R}} \end{aligned}$$

Definition

[38] A credibility distribution \(\Phi _A\) of a fuzzy variable A is called regular if it is a continuous and strictly increasing function w.r.t x such that\(0<\Phi _A<1\) and if \(lim_{x\rightarrow -\infty }\Phi _A=0,\) and \(lim_{x\rightarrow \infty }\Phi _A=1.\)

Definition

[38] Let A be a fuzzy variable with a regular credibility distribution \(\Phi _A\), then the inverse function \(\Phi _A^{-1}\) is called the inverse credibility distribution of A.

3 Construction of semi-elliptic fuzzy variable

Let’s consider the general equation of the horizontal ellipse centred at (a, b) is

$$\begin{aligned} \dfrac{(x-a)^2}{h^2}+\dfrac{(y-b)^2}{k^2}=1 \end{aligned}$$

To construct a normal semi-circular fuzzy variable (SEFV), it is needed to consider \(b=0\) and \(k=1\). i.e.,

$$\begin{aligned} \dfrac{(x-a)^2}{h^2}+y^2=1 \end{aligned}$$

Thus, the required membership function of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} \mu _A(x)=\sqrt{1-\dfrac{(x-a)^2}{h^2}}, a-h\le x\le a+h \end{aligned}$$

where a indicates the mean/core of the fuzzy variable while h determines width of the fuzzy variable A.

The \(\alpha\)-cut of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} ^\alpha A = \left[ a-h\sqrt{1-\alpha ^2},a+h\sqrt{1-\alpha ^2}\right] \end{aligned}$$

Suppose \(A=S_E(10,7)\) is SEFV representing a uncertain variable. The MF of the SEFV A is

$$\begin{aligned} \mu _A(x)=\sqrt{1-\dfrac{(x-10)^2}{49}}, 3\le x\le 17 \end{aligned}$$

The graphical representation of A is depicted in Fig. 1.

Fig. 1
figure 1

The SEFV \(A=S_E(10,7)\)

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4 Possibility, necessity and credibility measures of SEFV

In this section,the possibility measure, necessity measure and credibility measure of the SEFV are derived.

4.1 Possibility measures of SEFV

Suppose \(A=S_E(a,h)\) is a SEFV.

Then, the possibility measure of (\(A\le x\)) and (\(A\ge x\)) are respectively

$$\begin{aligned} Pos(A \le x)= & {} \left\{ \begin{array}{ll} \,\,\,1, &{}\,\,{\text{if}} \,\,x \ge a, \\ \sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, x<a, \\ \end{array} \right. \\ Pos(A \ge x)= & {} \left\{ \begin{array}{ll} \,\,\,1, &{}\,\,{\text{ if }} \,\, x \le a, \\ \sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, x>a, \\ \end{array} \right. \end{aligned}$$

The possibility measure of the SEFV \(A=S_E(10,7)\) for (\(A\le x\)) and (\(A\ge x\)) are depicted in Figs. 2 and 3 repectively.

Fig. 2
figure 2

The possibility measure of SEFV \(A=S_E(10,7)\) for (\(A\le x\))

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Fig. 3
figure 3

The possibility measure of SEFV \(A=S_E(10,7)\) for (\(A\ge x\))

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4.2 Necessity measures of SEFV

Suppose \(A=S_E(a,h)\) is a SEFV.

Then, the necessity measure of \(A\le x\) and \(A\ge x\) are respectively

$$\begin{aligned} Nec(A \le x)= & {} \left\{ \begin{array}{ll} \,\,\,0, &{}\,\,{\text{ if }} \,\,x \le a, \\ \sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, x>a, \\ \end{array} \right. \\ Nec(A \ge x)= & {} \left\{ \begin{array}{ll} \,\,\,0, &{}\,\,{\text{ if }} \,\, x \ge a, \\ \sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, x<a, \\ \end{array} \right. \end{aligned}$$

The necessity measure of the SEFV \(A=S_E(10,7)\) for (\(A\le x)\) and (\(A\ge x)\) are depicted in Figs. 4 and 5 repectively.

Fig. 4
figure 4

The necessity measure of SEFV \(A=S_E(10,7)\) for (\(A\le x)\)

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Fig. 5
figure 5

The necessity measure of SEFV \(A=S_E(10,7)\) for (\(A\ge x\))

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4.3 Credibility measures of SEFV

Suppose \(A=S_E(a,h)\) is a SEFV.

Then, the credibility measure of \(A\le x\) and \(A\ge x\) are respectively

$$\begin{aligned} Cr(A \le x)= & {} \left\{ \begin{array}{ll} \,\,\,\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\,a-h\le x \le a, \\ 1-\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, a\le x \le a+h, \\ \end{array} \right. \\ Cr(A \ge x)= & {} \left\{ \begin{array}{ll} 1-\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, a-h\le x \le a, \\ \,\,\,\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, a\le x \le a+h, \\ \end{array} \right. \end{aligned}$$

The necessity measure of the SEFV \(A=S_E(10,7)\) for (\(A\le x\)) and (\(A\ge x\)) are depicted in Figs. 6 and 7 repectively.

Fig. 6
figure 6

The credibility measure of SEFV \(A=S_E(10,7)\) for (\(A\le x\))

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Fig. 7
figure 7

The credibility measure of SEFV \(A=S_E(10,7)\) for (\(A\ge x\))

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4.4 Credibility distribution of SEFV

As the credibility distribution \(\Phi _A:{\mathbb {R}}\rightarrow [0,1]\) of a fuzzy variable A is defined as \(\Phi _A(x)=Cr \{\theta \in \Theta :\zeta (\theta )\le x\}\).

Hence, the credibility distribution of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} \Phi _A(x)=\left\{ \begin{array}{ll} \,\,\,\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\,a-h\le x \le a, \\ 1-\dfrac{1}{2}\sqrt{1-\dfrac{(x-a)^2}{h^2}}, &{}\,\,{\text{ if }} \,\, a\le x \le a+h, \\ \end{array} \right. \end{aligned}$$

The credibility distribution function of the SEFV \(A=S_E(10,7)\) is

$$\begin{aligned} \Phi _A(x)=\left\{ \begin{array}{ll} \,\,\,\dfrac{1}{2}\sqrt{1-\dfrac{(x-10)^2}{49}}, &{}\,\,{\text{ if }} \,\,3\le x \le 10, \\ 1-\dfrac{1}{2}\sqrt{1-\dfrac{(x-10)^2}{49}}, &{}\,\,{\text{ if }} \,\, 10\le x \le 17, \\ \end{array} \right. \end{aligned}$$

4.5 Inverse credibility distribution (ICD) of SEFV

The credibility distribution of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} \Phi ^{-1}(\alpha )==\left\{ \begin{array}{ll} a-h \sqrt{1-4\alpha ^2}, &{}\,\,{\text{ if }} \,\,0\le \alpha \le 0.5, \\ a+h \sqrt{1-4(1-\alpha )^2}, &{}\,\,{\text{ if }} \,\, \,\,0.5\le \alpha \le 1, \\ \end{array} \right. \end{aligned}$$

5 Expected value

Using the idea of credibility distribution, Liu and Liu [34] provided the expected value of fuzzy variables. Zhou et al. [38] presented expected value of fuzzy variables via ICD.

5.1 Expected value via credibility distribution

If A is a fuzzy variable then the expected value of A in terms of credibility distribution is defined as [34]

$$\begin{aligned}&E(A)=\int _0^{\infty }Cr\{A\ge r\}dr-\int _{\infty }^0 Cr\{A\le r\}dr\\ \end{aligned}$$

Then, the expected value of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} E(A)&=\int _{0}^{a-h}{dr}+\int _{a-h}^{a}\dfrac{1}{2}\sqrt{1-\dfrac{(r-a)^2}{h^2}}dr\\&\quad + \, \int _{a}^{a+h}\left\{ 1- \dfrac{1}{2}\sqrt{1-\dfrac{(r-a)^2}{h^2}}\right\} dr\\&=(a-h)+h +\int _{a-h}^{a}\dfrac{1}{2}\sqrt{1-\dfrac{(r-a)^2}{h^2}}dr\\&\quad - \, \int _{a}^{a+h} \dfrac{1}{2}\sqrt{1-\dfrac{(r-a)^2}{h^2}}dr\\&=a+\dfrac{\pi h}{8} -\dfrac{\pi h}{8} \\&=a \end{aligned}$$

5.2 Expected value via ICD

If A is a SEFV then the expected value of A in terms of ICD is defined as [38]

$$\begin{aligned} E[A]=\int _0^1\Phi ^{-1}(\alpha )d\alpha \end{aligned}$$

Then, the expected value of the SEFV \(A=S_E(a,h)\) is

$$\begin{aligned} E(A)&=\int _0^{0.5}\left\{ a-h\sqrt{1-4\alpha ^2}\right\} d\alpha \\&\quad +\, \int _{0.5}^{1}\left\{ a+h\sqrt{1-4(1-\alpha )^2}\right\} d\alpha \\&=a -\frac{h}{2}\int _{0}^{1}\sqrt{1-t^2}dt- \frac{h}{2}\int _{1}^{0}\sqrt{1-t^2}dt\\&=a \end{aligned}$$

Thus, in both the approaches, it is obtained that the expected value of the SEFV A is \(e=a\).

Remark

If \(A=S_E(h_1,h_2)\) is a asymmetric SEFV where \(h_1\) and \(h_2\) are left spread and right spread of A. Then, the expected value of A is

$$\begin{aligned} E[A]=a+\frac{\pi }{8}(h_2-h_1). \end{aligned}$$

6 Variance

In this section, variance of SEFV is calculated in terms of regular credibility distribution.

If A is a fuzzy variable with expected value e, then the variance of A is defined as [36]

$$\begin{aligned} V[A]=E[(A-e)^2] \end{aligned}$$

It should be noted that if the expected value e of the fuzzy variable is finite, then the variance satisfies

$$\begin{aligned} V[A]=E[(A-e)^2]=\int _0^{+\infty }Cr\left\{ (A-e)^2\ge r\right\} dr \end{aligned}$$

6.1 Variance of a SEFV

To evaluate variance V[A] of a SEFV A, it is needed to calculate the MF of \((A-e)^2\) first and to find MF of SEFV \(A=S_E(a,h)\), \(\alpha -\)cut technique is explored here.It is already obtained that the expected valued of SEFV is \(e=a\)

The \(\alpha\)-cut of the SEFV \(A=S_E(a,h)\) is \(^\alpha A = [a-h\sqrt{1-\alpha ^2},a+h\sqrt{1-\alpha ^2}]\).

The procedure is presented below.

$$\begin{aligned}&(^{\alpha }A-a)^2\\&\quad =\left[ (a-h\sqrt{1-\alpha ^2}-a)^2,(a+h\sqrt{1-\alpha ^2}-a)^2\right] \\&\quad =\left[ (-h\sqrt{1-\alpha ^2})^2,(h\sqrt{1-\alpha ^2})^2\right] \\&\quad =\left[ h^2(1-\alpha ^2),(h^2(1-\alpha ^2\right] \end{aligned}$$

Now, taking \(x=h^2(1-\alpha ^2)\) gives \(\alpha =\sqrt{1-\dfrac{x}{h^2}}\), \(0 \le x\le h^2\).

Thus, the MF of \((A-e)^2\) is

$$\begin{aligned} \mu _{(A-e)^2} (x)=\sqrt{1-\dfrac{x}{h^2}},0 \le x\le h^2. \end{aligned}$$

Since \(Cr\{(A-e)^2<r\}= \dfrac{1}{2}\{\underset{x<r}{Sup}\mu _{(A-e)^2}(x)+1 -\underset{x\ge r}{sup}\mu _{(A-e)^2}(x)\}\)

$$\begin{aligned} \therefore\, Cr\{(A-e)^2<r\}= \dfrac{1}{2}\left\{ 1+1-\sqrt{1-\dfrac{x}{h^2}}\right\} ,0 \le x\le h^2. \end{aligned}$$

Again, \(Cr\{(A-e)^2\ge r\}=1-Cr\{(A-e)^2<r\}\). Hence, \(Cr\{(A-e)^2\ge r\}={\left\{ \begin{array}{ll}\\ 1-Cr\{(A-e)^2<r\},\quad 0 \le x\le h^2\\ 0, \quad r>h^2 \end{array}\right. }.\) Then, the variance of SEFV \(A=S_E(a,h)\) is

$$\begin{aligned}&V[A]\\&\quad =\int _0^\infty {Cr\{(A-e)^2\ge r\}}dr\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\\&\quad =\int _0^{h^2}\dfrac{1}{2}\sqrt{1-\dfrac{r}{h^2}}dr\\&\quad =\dfrac{h^2}{3} \end{aligned}$$

For example, the variance of the SEFV \(A=S_E(10,7)\) is \(\dfrac{7^2}{3}=16.33\).

Remark

If the width h of a SEFV is unit then SEFV will represent a semi-circular fuzzy variable (SCFV). Then, it can be derived that for all SCFV the expected value is 0.33.

7 Rational upper bound of the variance

Yi et al. [39] derived the concept of rational upper bounded of the variance (RUBV) along with some definitions and results in terms of credibility distribution and ICD.

Definition

Let A be a fuzzy variable with credibility distribution \(\Phi\) and finite expected value e. The RUBV is defined as [39]

$$\begin{aligned} \overline{V}[A]=\int _0^\infty (1-\Phi (e+\sqrt{x})\Phi (e-\sqrt{x})) \end{aligned}$$

Definition

Let A be a fuzzy variable with credibility distribution \(\Phi\). If the expected value is e, then RUBV is evaluated as [39]

$$\begin{aligned} \overline{V}[A]=\int _0^1(\Phi ^{-1}(\alpha )-e)^2d\alpha \end{aligned}$$

Corollary

The RUBV of SEFV depends on the width h and it is \(\dfrac{2h^2}{3}.\)

Consider the SEFV \(A=S_E(a,h)\).

$$\begin{aligned}&\int _0^{0.5}h^2(1-4\alpha ^2)d\alpha +\int _{0.5}^1 h^2(1-4(1-\alpha )^2)d\alpha \\&\quad = h^2 + h^2\int _0^{0.5}4\alpha ^2d\alpha -h^2int_{0.5}^14(1-\alpha )^2d\alpha \\&\quad = h^2-\dfrac{h^2}{2}\int _0^1t^2dt+ \dfrac{h^2}{2}\int _1^0t^2dt\\&\quad = h^2-h^2\int _0^1t^2dt\\&\quad = h^2-\dfrac{h^2}{3}\\&\quad =\dfrac{2h^2}{3} \end{aligned}$$

Corollary

Let \(A=S_E(a,h)\) be SEFV. Then, \(\overline{V}[A]=2V[A]\).

Remark

The SCFV also satisfies the above corollary and it is also observed that for all SCFV, the RUBV is 0.66.

Corollary

Suppose \(A=S_E(a_1,h_1)\) and \(B=S_E(a_2,h_2)\) are two SRFV.

Then, \(\overline{V}[A+B]\le 2(\overline{V}[A]+\overline{V}[B]\).

For the two SEFV A and B, the ICV of \(A+B\) is

$$\begin{aligned}&\Phi ^{-1}(\alpha )\\&\quad =\Phi _A^{-1}(\alpha )+\Phi _B^{-1}(\alpha )\\&\quad ={\left\{ \begin{array}{ll} (a_1+b_1)-(h_1+h_2)\sqrt{1-4\alpha ^2},\alpha \le 0.5\\ (a_1+b_1)+(h_1+h_2)\sqrt{1-4(1-\alpha )^2},\alpha >0.5\\ \end{array}\right. } \end{aligned}$$

Now,

$$\begin{aligned}&\overline{V}[A+B]\\&\quad =\int _0^{0.5}[(a_1+b_1)-(h_1+h_2)\sqrt{1-4\alpha ^2}d\alpha \\&\qquad +\, \int _{0.5}^1(a_1+b_1)+(h_1+h_2)\sqrt{1-4(1-\alpha )^2}d\alpha \\&\quad =\dfrac{2(h_1+h_2)^2}{3}. \end{aligned}$$

Again, \(\overline{V}[A]=\dfrac{2(h_1)^2}{3}\) and \(\overline{V}[B]=\dfrac{2(h_2)^2}{3}\).

Since, \(\dfrac{2(h_1+h_2)^2}{3}\le \dfrac{2(h_1)^2}{3}+ \dfrac{2(h_2)^2}{3}.\)

Consequently, \(\overline{V}[A+B]\le 2(\overline{V}[A]+\overline{V}[B]\).

Corollary

Suppose \(A=S_E(a_1,h_1)\) and \(B=S_E(a_2,h_2)\) are two SRFVs.

Then,\(\sqrt{\overline{V}[A+B]}= \sqrt{(\overline{V}[A]}+\sqrt{\overline{V}[B]}\).

Since \(\sqrt{\overline{V}[A+B]}=\sqrt{\dfrac{2(h_1+h_2)^2}{3}}\) and \(\sqrt{\overline{V}[A]}=\sqrt{\dfrac{2h_1^2}{3}}\), \(\sqrt{\overline{V}[B]}=\sqrt{\dfrac{2h_2^2}{3}}\).

Thus, \(\sqrt{\overline{V}[A+B]}= \sqrt{(\overline{V}[A]}+\sqrt{\overline{V}[B]}\).

8 Arithmetic on SEFVs

In this section, basic operations on SEFVs are reviewed and adopted from [41].

Suppose \(A=S_E(a,h)\) and \(B=S_E(b,k)\) are two SEFVs defined on a universe of discourse X.

8.1 Addition

The membership function of \(A+B\) is

$$\begin{aligned} \mu _{(A+B)}&=\sqrt{1-{\Big \{\frac{x-(a+b)}{h_{}+k_{}}\Big \}}^{2}},\\&\quad x\in [(a-h_{})+(b-k_{}), \,(a+h_{})+(b+k_{})] \end{aligned}$$

8.2 Subtraction

The membership function of \(A-B\) is

$$\begin{aligned} \mu _{(A-B)}&=\sqrt{1-{\Big \{\dfrac{x-(a-b)}{h_{}+k_{}}\Big \}}^{2}},\\&\quad x\in [(a-h_{})-(b+k_{}),\,\, (a+h_{})-(b-k_{})] \end{aligned}$$

8.3 Multiplication

The membership function of AB is

$$\begin{aligned} \mu _{AB}(x)&=\sqrt{1-\Big \{\frac{(ak_{}+bh_{})-\sqrt{(ak_{}+bh_{})^{2}-4h_{}k_{}(ab-x)}}{2h_{}k_{}}\Big \}^{}},\\&\quad x \in (a-h_{})(b-k_{})\le x \le (a+h_{})(b+k_{}) \end{aligned}$$

8.4 Division

The membership function of A / B is

$$\begin{aligned} \mu _{\frac{A}{B}}(x)&=\sqrt{1-\Big \{\frac{a-xb}{h_{}+xk_{}}\Big \}^{}},\,\frac{a-h_{}}{b+k_{}}\le x \le \frac{a+h_{}}{b-k_{}} \end{aligned}$$

9 Rank of SEFVs

In this section, ranking of two SEFVs are defined based on the expected value and variance of the SEFVs. Suppose \(A=S_E(a_1,h_1)\) and \(B=S_E(a_2,h_2)\) are two SEFVs.

Then rank of A and B are defined as

  1. I.

    \(A\le B\) if \(E[A]\le E[B]\).

  2. II.

    If \(E[A]= E[B]\) then

\(A\le B\) if \(V[A]\ge V[B]\)

Example 9.1

Suppose \(A = [-4, 0, 4]\) and \(B = [-2, 0, 2]\) are two fuzzy variables adopted from [42]. It is observed that the approaches [42,43,44,45,46,47] fail to compare the fuzzy variables. Reforming these fuzzy variables to SEFVs as \(A = S_E(0,4)\) and \(B = S_E(0,2)\) and applying the present approach it is obtained that \(A\le B\) which is consistent with human intuitions. Here, \(E[A]= E[B]= 0\), but \(V[A]= 5.33 \ge V[B]= 1.33\) and consequently, it can be adopted that \(A\le B\). A details comparison has been presented in Table 1.

Table 1 Ranking of fuzzy variables for example 8.1
Full size table

Example 9.2

Consider the fuzzy variables \(A=[0.2,0.5,0.8]\) and \(B=[0.4,0.5,0.6]\). The approaches [44, 47,48,49, 51] are not applicable to distinguish A and B while [52,53,54] and [50] produces illogical output. Here also the general or human intuition is that \(A\le B\). Applying the present approach we obtain \(E[A]= E[B]= 0.5\), but \(V[A]= 0.1 \ge V[B]= 0.033\) which gives \(A\le B\). A detail discussion is presented in Table 2.

Table 2 Ranking of fuzzy variables for example 8.2
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From the above analysis it can be opined that the present ranking approach has the capability to overcome the drawbacks of the existing approaches.

10 Application of SEFVs

In this segment, application of SEFVs are performed in structural reliability analysis and medical diagnosis. For structural reliability arithmtic of SEFVs are taken into consideration while in medical diagnosis arithmetic as well as ranking of SEFVs are adopted.

10.1 Application in structural reliability

Depictions of the elements of structural analysis are more often SEFVs in some circumstances. In such circumstances structural failure can be evaluated using arithmetic of SEFVs. Consider the following problem of structural failure adopted from Dutta [3].

Example 10.1

Suppose a beam of height \(h=9 mm\), length \(L=1250 mm\) and the force density \(f=78.5\times 10^{-5} kN/mm^3\). The load w, breadth of the beam b and ultimate bending moment \(M_o\) are uncertain input variables represented by SEFVs where \(w=S_E(400,15)kN\), \(b=S_E(40,5)\) and \(M_o=S_E(2.05\times 10^5, 0.05\times 10^5) kN-mm.\) which is depicted in Fig. 8.

The limit state function is \(g(b,f,h,w,M_o,L)=M_o-\left( \dfrac{wL}{4}+\dfrac{fbhL^2}{8}\right)\).

It is needed to evaluate the structural failure of the beam.

Fig. 8
figure 8

Beam associated with its bending moment

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Applying the arithmetic of SEFVs on the problem the value of structural reliability or g is obtained as \(S_E(0.248\times 10^5,0.1658\times 10^5)\).

10.2 Application in medical diagnosis

It is observed that patient’s explanations, medical information even medical assessment process tainted with imprecision/vagueness/uncertainty. On the other hand, knowledge base correlating the symptom-disease relationship encompasses of ambiguity and uncertainty in medical assessment process. Accordingly to deal with such uncertainties FST is being adopted and became most demanding area in medical assessment process. Here, SEFVs are considered to represent uncertain information.

Consider the Patient-symptom and Symptom-disease relations presented in Tables 3 and 4, respectively.

Table 3 Patient-symptom relation
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Table 4 Symptom-disease relation
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Table 5 Patient-Disease relation
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Table 6 Crisp values of the patient-disease relation
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Using the multiplication and addition of SEFVs the resultant Patient-disease relation is evaluated and presented in Table 5. Then, ranking of SEFVs is adopted to obtain crisp values of resultant SEFVs and presented in Table 6. It should be noted that maximum value in each row indicates that the patient is likely to have the disease. Here, \(\{P_1, P_2, P_3\}\), \(\{S_1, S_2, S_3\}\) and \(\{D_1, D_2,D_3\}\) are the set of patients, symptoms and diseases, respectively. From Table 6, it is clear that the maximum value (the bold value) in 1st row is 59.0686 which associates patient \(P_1\) and disease \(D_2\). That is, patient \(P_1\) is likely to have the disease \(D_2\). Similarly, from 2nd and 3rd row (bold values in the Table 6) it can be opined that patient \(P_2\) is suffering from disease \(D_1\) and patient \(P_3\) is suffering from disease \(D_3\).

11 Conclusions

Uncertainty is an integral part of real world problems such as reliability assessment as well as medical diagnosis problems. To cope with uncertainty a handful number of fuzzy variables are demonstrated yet in literature. However, a special complicated fuzzy variable SEFV is not deliberated well and in this regard here SEFV has been introduced in terms of credibility theory first. Then, some important properties such as possibility measure, necessity measure, credibility measure, credibility distribution and ICD were presented. Afterwards, investigations on expected value of SEFV using credibility distribution and ICD along with variance and RUVB of SEFV have been performed and established relationship between them. Another important concept ranking of SEFVs is introduced based on expected value of SEFV and if it fails then variance of SEFVs concept has been utilized to evaluate order of SEFVs. Comparative numerical illustrations have been presented where results of existing methods and present method were compared and exhibited that present method smoothly over come the limitations of earlier methods. Finally, reliability analysis has been performed using arithmetic of SEFV while a medical diagnosis is performed using arithmetic and rank of SEFVs as well. The present model successfully solves both the problems which exhibits the novelty and applicability of the present model. However, the limitation of this present model is that it can’t work properly when asymmetric SEFVs come into picture. Therefore, as an extension of this work, asymmetric SEFV will be investigated.