Medical Diagnosis Based on SingleValued Neutrosophic Probabilistic Rough Multisets over Two Universes
Abstract
:1. Introduction
2. Preliminaries
2.1. SVNMs
 1.
 $A\oplus B=\left\{\u2329x,{T}_{A}^{i}\left(x\right)+{T}_{B}^{i}\left(x\right){T}_{A}^{i}\left(x\right){T}_{B}^{i}\left(x\right),{I}_{A}^{i}\left(x\right){I}_{B}^{i}\left(x\right),{F}_{A}^{i}\left(x\right){F}_{B}^{i}\left(x\right)\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$;
 2.
 $A\otimes B=\left\{\u2329x,{T}_{A}^{i}\left(x\right){T}_{B}^{i}\left(x\right),{I}_{A}^{i}\left(x\right)+{I}_{B}^{i}\left(x\right){I}_{A}^{i}\left(x\right){I}_{B}^{i}\left(x\right),{F}_{A}^{i}\left(x\right)+{F}_{B}^{i}\left(x\right){F}_{A}^{i}\left(x\right){F}_{B}^{i}\left(x\right)\u232a\right.\leftx\in U,\right.$$\left.i=1,2,\dots ,q\right\}$;
 3.
 the complement of A is represented by ${A}^{c}$ such that $\forall x\in U$,${A}^{c}=\left\{\u2329x,{F}_{A}^{i}\left(x\right),1{I}_{A}^{i}\left(x\right),{T}_{A}^{i}\left(x\right)\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$;
 4.
 the union of A and B is represented by $A\cup B$ such that $\forall x\in U$,$A\cup B=\left\{\u2329x,{T}_{A}^{i}\left(x\right)\vee {T}_{B}^{i}\left(x\right),{I}_{A}^{i}\left(x\right)\wedge {I}_{B}^{i}\left(x\right),{F}_{A}^{i}\left(x\right)\wedge {F}_{B}^{i}\left(x\right)\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$;
 5.
 the intersection of A and B is represented by $A\cap B$ such that $\forall x\in U$,$A\cap B=\left\{\u2329x,{T}_{A}^{i}\left(x\right)\wedge {T}_{B}^{i}\left(x\right),{I}_{A}^{i}\left(x\right)\vee {I}_{B}^{i}\left(x\right),{F}_{A}^{i}\left(x\right)\vee {F}_{B}^{i}\left(x\right)\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$.
2.2. PRSs
3. Probabilistic Rough Approximations of a SVNM under the Background of Two Universes
3.1. Relations on SVNMs Based on Two Universes and Some Operations
 1.
 $A\ominus B=\left\{\u2329x,\frac{{T}_{A}^{i}\left(x\right){T}_{B}^{i}\left(x\right)}{1{T}_{B}^{i}\left(x\right)},\frac{{I}_{A}^{i}\left(x\right)}{{I}_{B}^{i}\left(x\right)},\frac{{F}_{A}^{i}\left(x\right)}{{F}_{B}^{i}\left(x\right)}\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$, which is valid under the requirements $A\ge B$, ${T}_{B}^{i}\left(x\right)\ne 1$, ${I}_{B}^{i}\left(x\right)\ne 0$ and ${F}_{B}^{i}\left(x\right)\ne 0$;
 2.
 $A\oslash B=\left\{\u2329x,\frac{{T}_{A}^{i}\left(x\right)}{{T}_{B}^{i}\left(x\right)},\frac{{I}_{A}^{i}\left(x\right){I}_{B}^{i}\left(x\right)}{1{I}_{B}^{i}\left(x\right)},\frac{{F}_{A}^{i}\left(x\right){F}_{B}^{i}\left(x\right)}{1{F}_{B}^{i}\left(x\right)}\u232a\leftx\in U,i=1,2,\dots ,q\right.\right\}$, which is valid under the requirements $A\le B$, ${T}_{B}^{i}\left(x\right)\ne 0$, ${I}_{B}^{i}\left(x\right)\ne 1$ and ${F}_{B}^{i}\left(x\right)\ne 1$.
 1.
 If $s\left({x}_{1}\right)<s\left({x}_{2}\right)$, then ${x}_{1}<{x}_{2}$;
 2.
 If $s\left({x}_{1}\right)=s\left({x}_{2}\right)$, then ${x}_{1}={x}_{2}$;
 3.
 If $s\left({x}_{1}\right)>s\left({x}_{2}\right)$, then ${x}_{1}>{x}_{2}$.
3.2. SVNRMs over Two Universes
3.3. PRSVNMs over Two Universes
3.4. SVNPRMs over Two Universes
 1.
 $A\subseteq B\Rightarrow {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(B\right)$, ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(B\right)$;
 2.
 ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(\varnothing \right)=\varnothing $, ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(V\right)=U$;
 3.
 ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)$;
 4.
 ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\cap B\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\cap {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(B\right)$, ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\cup B\right)\supseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)\cup {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(B\right)$;
 5.
 ${\alpha}_{1}\le {\alpha}_{2}\Rightarrow {\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{2}}\left(A\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{1}}\left(A\right)$, ${\beta}_{1}\le {\beta}_{2}\Rightarrow {\overline{SVNMR}}_{{}_{P}}^{{\beta}_{2}}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{{\beta}_{1}}\left(A\right)$.
4. Medical Diagnosis Based on SVNPRMs over Two Universes
4.1. Medical Diagnosis Model
 (P)
 If ${x}_{i}\in PO{S}_{SVNMR}\left(A,\alpha ,\beta \right)$, $i=1,2,\dots ,m$, then ${x}_{i}$ is the determined diagnostic conclusion;
 (N)
 If ${x}_{i}\in NE{G}_{SVNMR}\left(A,\alpha ,\beta \right)$, $i=1,2,\dots ,m$, then ${x}_{i}$ is the excluded diagnostic conclusion;
 (B)
 If ${x}_{i}\in BN{D}_{SVNMR}\left(A,\alpha ,\beta \right)$, $i=1,2,\dots ,m$, then medical experts are unable to confirm whether ${x}_{i}$ is the determined or excluded diagnostic conclusion, they need more additional medical examinations to confirm the final diagnostic conclusion
4.2. Algorithm for Medical Diagnosis Model
Algorithm 1 Medical diagnosis based on SVNPRMs over two universes. 

4.3. An Illustrative Example
 (P)
 The patient is suffering from viral fever, medical experts need to pay close attention to the diagnosis;
 (N)
 The same patient shows no signs of having throat disease, which does not need close attention at the current stage;
 (B)
 Medical experts are unable to confirm whether the considered patient is suffering from tuberculosis and typhoid or not due to insufficient diagnostic information, they might organize an expert consultation to confirm the determined diagnostic conclusion at a later stage.
4.4. Comparative Analysis
4.4.1. Comparison with Other Approaches in Literature [26]
4.4.2. Comparison with Other Approaches in Literature [28]
 (P)
 The customer is suggested to purchase the third car;
 (N)
 The same customer is not suggested to by the first car at present;
 (B)
 The same customer is not sure whether the second car and the forth car are the ideal selections, he or she might collect some additional information to make a final conclusion at a later stage.
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
 Since $A\subseteq B$, according to Definition 14, we have${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)=\left\{\frac{{\sum}_{y\in V}A\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge \alpha \leftx\in U,y\in V\right.\right\}\subseteq $$\left\{\frac{{\sum}_{y\in V}B\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge \alpha \leftx\in U,y\in V\right.\right\}={\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(B\right)$.Thus we obtain $A\subseteq B\Rightarrow {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(B\right)$. Similarly, we could also obtain ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(B\right)$.
 ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(\varnothing \right)=\left\{\frac{{\sum}_{y\in V}\varnothing R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge \alpha \leftx\in U,y\in V\right.\right\}=\varnothing $,${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(V\right)=\left\{\frac{{\sum}_{y\in V}VR\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}>\beta \leftx\in U,y\in V\right.\right\}=U$.Hence ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(\varnothing \right)=\varnothing $ and ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(V\right)=U$ could be obtained.
 Since $0\le \beta \le \alpha \le 1$, it is not difficult to obtain${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)=\left\{\frac{{\sum}_{y\in V}A\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge \alpha \leftx\in U,y\in V\right.\right\}\subseteq $$\left\{\frac{{\sum}_{y\in V}B\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}>\beta \leftx\in U,y\in V\right.\right\}={\overline{SVNMR}}_{{}_{P}}^{\beta}\left(B\right)$.Therefore, ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)$ could be obtained.
 If ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\cap B\right)$ holds, we have $P\left(\left(A\cap B\right)\leftR\left(x,y\right)\right.\right)\ge \alpha $, then it is not difficult to obtain$\alpha \le \frac{{\sum}_{y\in V}\left(A\cap B\right)\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\le \frac{{\sum}_{y\in V}A\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}$ and $\alpha \le \frac{{\sum}_{y\in V}\left(A\cap B\right)\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\le \frac{{\sum}_{y\in V}B\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}$.Hence we obtain ${\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\cap B\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(A\right)\cap {\underline{SVNMR}}_{{}_{P}}^{\alpha}\left(B\right)$. In an identical fashion, ${\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\cup B\right)\supseteq {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(A\right)\cup {\overline{SVNMR}}_{{}_{P}}^{\beta}\left(B\right)$ could also be obtained.
 Since ${\alpha}_{1}\le {\alpha}_{2}$, we have${\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{2}}\left(A\right)=\left\{\frac{{\sum}_{y\in V}A\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge {\alpha}_{2}\leftx\in U,y\in V\right.\right\}\subseteq $$\left\{\frac{{\sum}_{y\in V}A\left(y\right)R\left(x,y\right)}{{\sum}_{y\in V}R\left(x,y\right)}\ge {\alpha}_{1}\leftx\in U,y\in V\right.\right\}={\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{1}}\left(A\right)$.Hence we have ${\alpha}_{1}\le {\alpha}_{2}\Rightarrow {\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{2}}\left(A\right)\subseteq {\underline{SVNMR}}_{{}_{P}}^{{\alpha}_{1}}\left(A\right)$, and ${\beta}_{1}\le {\beta}_{2}\Rightarrow {\overline{SVNMR}}_{{}_{P}}^{{\beta}_{2}}\left(A\right)\subseteq {\overline{SVNMR}}_{{}_{P}}^{{\beta}_{1}}\left(A\right)$ could be proved in a similar way.
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R  ${\mathit{y}}_{1}$  ${\mathit{y}}_{2}$  ${\mathit{y}}_{3}$  ${\mathit{y}}_{4}$  ${\mathit{y}}_{5}$ 

${x}_{1}$  $\u2329\left(0.8\right),\left(0.1\right),\left(0.1\right)\u232a$  $\u2329\left(0.2\right),\left(0.7\right),\left(0.1\right)\u232a$  $\u2329\left(0.3\right),\left(0.5\right),\left(0.2\right)\u232a$  $\u2329\left(0.5\right),\left(0.3\right),\left(0.2\right)\u232a$  $\u2329\left(0.5\right),\left(0.4\right),\left(0.1\right)\u232a$ 
${x}_{2}$  $\u2329\left(0.2\right),\left(0.7\right),\left(0.1\right)\u232a$  $\u2329\left(0.9\right),\left(0.0\right),\left(0.1\right)\u232a$  $\u2329\left(0.7\right),\left(0.2\right),\left(0.1\right)\u232a$  $\u2329\left(0.6\right),\left(0.3\right),\left(0.1\right)\u232a$  $\u2329\left(0.7\right),\left(0.2\right),\left(0.1\right)\u232a$ 
${x}_{3}$  $\u2329\left(0.5\right),\left(0.3\right),\left(0.2\right)\u232a$  $\u2329\left(0.3\right),\left(0.5\right),\left(0.2\right)\u232a$  $\u2329\left(0.2\right),\left(0.7\right),\left(0.1\right)\u232a$  $\u2329\left(0.2\right),\left(0.6\right),\left(0.2\right)\u232a$  $\u2329\left(0.4\right),\left(0.4\right),\left(0.2\right)\u232a$ 
${x}_{4}$  $\u2329\left(0.1\right),\left(0.7\right),\left(0.2\right)\u232a$  $\u2329\left(0.3\right),\left(0.6\right),\left(0.1\right)\u232a$  $\u2329\left(0.8\right),\left(0.1\right),\left(0.1\right)\u232a$  $\u2329\left(0.1\right),\left(0.8\right),\left(0.1\right)\u232a$  $\u2329\left(0.1\right),\left(0.8\right),\left(0.1\right)\u232a$ 
R  ${\mathit{y}}_{1}$  ${\mathit{y}}_{2}$  ${\mathit{y}}_{3}$  ${\mathit{y}}_{4}$ 

${x}_{1}$  $\u2329\left(0.7,0.5\right),\left(0.7,0.3\right),\left(0.6,0.2\right)\u232a$  $\u2329\left(0.5\right),\left(0.4\right),\left(0.4\right)\u232a$  $\u2329\left(0.8,0.7\right),\left(0.7,0.7\right),\left(0.6,0.5\right)\u232a$  $\u2329\left(0.5,0.1\right),\left(0.5,0.2\right),\left(0.8,0.7\right)\u232a$ 
${x}_{2}$  $\u2329\left(0.9,0.7\right),\left(0.7,0.7\right),\left(0.5,0.1\right)\u232a$  $\u2329\left(0.8\right),\left(0.7\right),\left(0.6\right)\u232a$  $\u2329\left(0.9,0.9\right),\left(0.6,0.6\right),\left(0.4,0.4\right)\u232a$  $\u2329\left(0.5,0.5\right),\left(0.2,0.1\right),\left(0.9,0.7\right)\u232a$ 
${x}_{3}$  $\u2329\left(0.6,0.3\right),\left(0.4,0.3\right),\left(0.7,0.2\right)\u232a$  $\u2329\left(0.2\right),\left(0.2\right),\left(0.2\right)\u232a$  $\u2329\left(0.9,0.6\right),\left(0.5,0.5\right),\left(0.5,0.2\right)\u232a$  $\u2329\left(0.7,0.4\right),\left(0.5,0.2\right),\left(0.3,0.2\right)\u232a$ 
${x}_{4}$  $\u2329\left(0.9,0.8\right),\left(0.7,0.6\right),\left(0.2,0.1\right)\u232a$  $\u2329\left(0.5\right),\left(0.3\right),\left(0.2\right)\u232a$  $\u2329\left(0.5,0.1\right),\left(0.7,0.4\right),\left(0.5,0.2\right)\u232a$  $\u2329\left(0.8,0.8\right),\left(0.4,0.4\right),\left(0.2,0.2\right)\u232a$ 
Different Methods  Ranking Results of Alternatives  The Best Alternative 

Method 1 based on similarity measures in [26]  ${x}_{1}\succ {x}_{3}\succ {x}_{2}\succ {x}_{4}$  ${x}_{1}$ 
The proposed method  ${x}_{1}\succ {x}_{2}\succ {x}_{3}\succ {x}_{4}$  ${x}_{1}$ 
Different Methods  Ranking Results of Alternatives  The Best Alternative 

Method 2 based on cosine measures in [28]  ${x}_{3}\succ {x}_{4}\succ {x}_{1}\succ {x}_{2}$  ${x}_{3}$ 
The proposed method  ${x}_{3}\succ {x}_{2}\succ {x}_{4}\succ {x}_{1}$  ${x}_{3}$ 
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Zhang, C.; Li, D.; Broumi, S.; Sangaiah, A.K. Medical Diagnosis Based on SingleValued Neutrosophic Probabilistic Rough Multisets over Two Universes. Symmetry 2018, 10, 213. https://doi.org/10.3390/sym10060213
Zhang C, Li D, Broumi S, Sangaiah AK. Medical Diagnosis Based on SingleValued Neutrosophic Probabilistic Rough Multisets over Two Universes. Symmetry. 2018; 10(6):213. https://doi.org/10.3390/sym10060213
Chicago/Turabian StyleZhang, Chao, Deyu Li, Said Broumi, and Arun Kumar Sangaiah. 2018. "Medical Diagnosis Based on SingleValued Neutrosophic Probabilistic Rough Multisets over Two Universes" Symmetry 10, no. 6: 213. https://doi.org/10.3390/sym10060213