Mathematics of uncertainty: an exploration on semi-elliptic fuzzy variable and its properties
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Abstract
Uncertainty is more or less encountered in industrial and medical systems as well. Uncertainty theory is an upgraded theory which comprises of possibility measure, necessity measure and credibility measure plays significant role in modelling uncertainty. In connection with uncertainty modeling, a special and intricate fuzzy variable viz., semi-elliptic fuzzy variable (SEFV) is thrashed out here. Subsequently, an attempt has been made to derive possibility, necessity and credibility measure of the SEFV first. Later, some other properties such as expected value, variance, rational upper bound etc are presented and based on that mean and variance ranking of SEFVs are proposed. Afterwards reliability analysis and medical diagnosis cases are carried out which exhibit the efficiency and novelty of the derived SEFV. In this work done, it is observed that the present work has the capability to resolve problems under uncertain complex situations.
Keywords
Uncertainty Fuzzy variable Semi-elliptic fuzzy variable Possibility measure Necessity measure Credibility distribution1 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
Definition
Definition
Definition
Definition
\(Cr(\Theta )=1\)
\(Cr(A)\le Cr(B)\) 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 \(\underset{i}{Sup}Cr\{A_i\}<0.5\)
Definition
Definition
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
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.
4.2 Necessity measures of SEFV
Suppose \(A=S_E(a,h)\) is a SEFV.
4.3 Credibility measures of SEFV
Suppose \(A=S_E(a,h)\) is a SEFV.
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 \(\Phi _A(x)=Cr \{\theta \in \Theta :\zeta (\theta )\le x\}\).
4.5 Inverse credibility distribution (ICD) of SEFV
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
5.2 Expected value via ICD
Remark
6 Variance
In this section, variance of SEFV is calculated in terms of regular credibility distribution.
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 \(^\alpha A = [a-h\sqrt{1-\alpha ^2},a+h\sqrt{1-\alpha ^2}]\).
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
Definition
Corollary
The RUBV of SEFV depends on the width h and it is \(\dfrac{2h^2}{3}.\)
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, \(\overline{V}[A+B]\le 2(\overline{V}[A]+\overline{V}[B]\).
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
8.2 Subtraction
8.3 Multiplication
8.4 Division
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.
- I.
\(A\le B\) if \(E[A]\le E[B]\).
- 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.
Ranking of fuzzy variables for example 8.1
Approaches | A | B | Result |
---|---|---|---|
Abbasbandy et al. [42] | 0 | 0 | \(A\sim B\) |
Wang [43] | 0 | 0 | \(A\sim B\) |
Asady [44] | 0.9 | 0.9 | \(A\sim B\) |
Asady [45] | 0 | 0 | \(A\sim B\) |
Asady [46] | 0.444 | 0.444 | \(A\sim B\) |
Abbasbandy and Hajjari [47] | 0.117 | 0.074 | \(A\succ B\) |
Present approach | \(E[A]=0.5\) | \(E[B]=0.5\) | |
\(V[A]=5.33\) | V[B]=1.33 | \(A\le B\) |
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.
Ranking of fuzzy variables for example 8.2
Approaches | Decision-level | A | B | Result |
---|---|---|---|---|
Yager [48] | 0.5 | 0.5 | \(A\sim B\) | |
Abbhasbandy and Hajjari [47] | 0.5 | 0.5 | \(A\sim B\) | |
Asady [44] | 0.374 | 0.374 | \(A\sim B\) | |
Chen and Sanguansat [51] | 0.5 | 0.5 | \(A\sim B\) | |
Chen et al. [49] | 0.444 | 0.444 | \(A\sim B\) | |
\(\alpha =1\) | 0.117 | 0.074 | \(A\succ B\) | |
Vincent and Dat [54] | \(\alpha =1\) | 0.45 | 0.35 | \(A\succ B\) |
Rezvani [50] | 0.078 | 0.062 | \(A\succ B\) | |
Present Approach | \(E[A]=0.5\) | \(E[B]=0.5\) | ||
\(V[A]=0.1\) | V[B]=0.033 | \(A\le B\) |
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.
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.
Patient-symptom relation
\(R_1\) | \(S_1\) | \(S_2\) | \(S_3\) |
---|---|---|---|
\(P_1\) | \(S_E(3,1)\) | \(S_E(6,1)\) | \(S_E(2,1)\) |
\(P_2\) | \(S_E(7,1)\) | \(S_E(2,1)\) | \(S_E(4,2)\) |
\(P_3\) | \(S_E(2,1)\) | \(S_E(3,1)\) | \(S_E(7,2)\) |
Symptom-disease relation
\(R_2\) | \(D_1\) | \(D_2\) | \(D_3\) |
---|---|---|---|
\(S_1\) | \(S_E(6,2)\) | \(S_E(4,2)\) | \(S_E(3,1)\) |
\(S_2\) | \(S_E(2,1)\) | \(S_E(6,1)\) | \(S_E(2,1)\) |
\(S_3\) | \(S_E(3,1)\) | \(S_E(2,1)\) | \(S_E(7,2)\) |
Patient-Disease relation
R | \(D_1\) | \(D_2\) | \(D_3\) |
---|---|---|---|
\(P_1\) | \(S_E(36,17,40)\) | \(S_E(52,20,38)\) | \(S_E(35,19,33)\) |
\(P_2\) | \(S_E(58,29,35)\) | \(S_E(48,29,28)\) | \(S_E(53,30,28)\) |
\(P_3\) | \(S_E(39,23,29)\) | \(S_E(40,23,27)\) | \(S_E(61,32,27)\) |
Crisp values of the patient-disease relation
R | \(D_1\) | \(D_2\) | \(D_3\) |
---|---|---|---|
\(P_1\) | 45.0321 | 59.0686 | 40.4978 |
\(P_2\) | 60.3562 | 47.6073 | 52.2146 |
\(P_3\) | 41.3562 | 41.5708 | 59.0365 |
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.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human or animal subjects.
Supplementary material
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