Abstract
This paper mainly aims to introduce Quintuple Implication Principle (QIP) on interval-valued intuitionistic fuzzy sets (IVIFSs). Firstly, some algebraic properties of a class of interval-valued intuitionistic triangular norms are discussed in detail. In particular, a unified expression of residual interval-valued intuitionistic fuzzy implications generated by left-continuous triangular norms is presented. Secondly, Triple Implication Principles (TIPs) of both interval-valued intuitionistic fuzzy modus ponens (IVIFMP) and fuzzy modus tollens (IVIFMT) based on residual interval-valued intuitionistic fuzzy implications are analyzed. It is shown that the TIP solution of IVIFMP is recoverable, and the TIP solution of IVIFMT is only weakly local recoverable. Moreover, it sees by an illustrated example that the TIP method sometimes makes the computed solutions for IVIFMP and IVIFMT meaningless or misleading. To avoid the above shortcoming and enhance the recovery property of TIP solution of IVIFMT, QIP and \(\alpha \)-QIP for IVIFMP and IVIFMT are investigated and the corresponding expressions of solutions of them are also given, respectively. In addition, the QIP methods for IVIFMP and IVIFMT are recoverable and sound. Finally, QIP solutions of IVIFMP for multiple fuzzy rules are provided. An application example for medical diagnosis is given to illustrate the feasibility and effectiveness of the QIP of IVIFMP.
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Acknowledgements
The authors would like to thank the anonymous referees for their careful reading of this paper and for the valuable comments and suggestions which improved the quality of this paper. This work is supported by National Natural Science Foundation of China (Grant No. 11401495), Fund of China Scholarship Council (No. 201708515152) and Graduate Educational Reform Project of Southwest Petroleum University (No.18YJZD08).
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Appendices
Appendix 1
The proof of Proposition 7 is shown as follows:
-
(1)
Since \((\otimes _{SL^*},\rightarrow _{SL^*})\) is an interval-valued intuitionistic adjoint pair, then \(\gamma \rightarrow _{SL^*} \eta =\mathbf{1 }^*\) if and only if \(\mathbf{1 }^*\otimes _{SL^*}\gamma \le \eta \) if and only if \(\gamma \le \eta \).
-
(2)
\(\eta \le \gamma \rightarrow _{SL^*}\beta \) if and only if \(\eta \otimes _{SL^*}\gamma \le \beta \) if and only if \(\gamma \otimes _{SL^*}\eta \le \beta \) if and only if \(\gamma \le \eta \rightarrow _{SL^*} \beta \).
-
(3)
\(\forall \alpha \in SL^*\), \(\alpha \le \gamma \rightarrow _{SL^*}(\eta \rightarrow _{SL^*}\beta )\) if and only if \(\alpha \otimes _{SL^*}\gamma \le \eta \rightarrow _{SL^*}\beta \) if and only if \(\alpha \otimes _{SL^*}\gamma \otimes _{SL^*}\eta \le \beta \) if and only if \(\alpha \otimes _{SL^*}\eta \le \gamma \rightarrow _{SL^*}\beta \) if and only if \(\alpha \le \eta \rightarrow _{SL^*} (\gamma \rightarrow _{SL^*}\beta )\), that is, \(\gamma \rightarrow _{SL^*}(\eta \rightarrow _{SL^*}\beta )=\eta \rightarrow _{SL^*}(\gamma \rightarrow _{SL^*}\beta )\).
-
(4)
\(\mathbf{1 }^*\rightarrow _{SL^*}\eta =\vee \{\gamma \in SL^*|\gamma \otimes _{SL^*}\mathbf{1 }^*\le \eta \}=\eta \).
-
(5)
\(\forall \alpha \in SL^*\), \(\alpha \le \gamma \rightarrow _{SL^*}(\wedge _{i\in I}\beta _i)\) if and only if \(\alpha \otimes _{SL^*}\gamma \le \wedge _{i\in I}\beta _i\) if and only if \(\forall i\in I\), \(\alpha \otimes _{SL^*}\gamma \le \beta _i\) if and only if \(\forall i\in I\), \(\alpha \le \gamma \rightarrow _{SL^*}\beta _i\) if and only if \(\alpha \le \wedge _{i\in I}(\gamma \rightarrow _{SL^*}\beta _i)\), that is, \(\gamma \rightarrow _{SL^*}(\wedge _{i\in I}\beta _i)=\wedge _{i\in I}(\gamma \rightarrow _{SL^*}\beta _i)\).
-
(6)
Since \(\otimes _{SL^*}\) is left continuous, then \(\forall \alpha \in SL^*\), \(\alpha \le (\vee _{i\in I}\gamma _i)\rightarrow _{SL^*} \beta \) if and only if \(\alpha \otimes _{SL^*}(\vee _{i\in I}\gamma _i)\le \beta \) if and only if \(\vee _{i\in I}(\alpha \otimes _{SL^*}\gamma _i)\le \beta \) if and only if \(\forall i\in I\),\(\alpha \otimes _{SL^*}\gamma _i\le \beta \) if and only if \(\forall i\in I\),\(\alpha \le \gamma _i \rightarrow _{SL^*}\beta \) if and only if \(\alpha \le \wedge _{i\in I}(\gamma _i\rightarrow _{SL^*}\beta )\), that is, \((\vee _{i\in I}\gamma _i)\rightarrow _{SL^*} \beta =\wedge _{i\in I}(\gamma _i\rightarrow _{SL^*}\beta )\).
-
(7)
Let \(\forall \alpha _1, \alpha _2\in SL^*\) and \(\alpha _1\le \alpha _2\), then \(\alpha _1\wedge \alpha _2=\alpha _1\), it follows from Proposition 7 (5) that \(\beta \rightarrow _{SL^*}\alpha _1=\beta \rightarrow _{SL^*}(\alpha _1\wedge \alpha _2)=(\beta \rightarrow _{SL^*}\alpha _1)\wedge (\beta \rightarrow _{SL^*}\alpha _2)\le \beta \rightarrow _{SL^*}\alpha _2\), that is, \(\rightarrow _{SL^*}\) is isotone in the second variable. Similarly, it follows from Proposition 7 (6) that \(\rightarrow _{SL^*}\) is antitone in the first variable.
Appendix 2
The proof of Theorem 2 is shown as follows:
Let \(\gamma =([a,b],[c,d])=\alpha \rightarrow _{SL^*}\beta \),
\(\eta _i=([e_i,f_i],[h_i,k_i])\in SL^*\). From Theorem 1, it follows that
We firstly compute the second element [c, d] of \(\gamma \), it could be given by the following form:
\([c,d]=\bigwedge \{[h_i,k_i]\mid c_2\le h_i\oplus c_1, d_2\le k_i\oplus d_1\}=\bigwedge \{[h_i,k_i]\mid [c_2,d_2]\le [h_i,k_i]\oplus _{SI}[c_1,d_1] \}=[c_2\ominus c_1,(c_2\ominus c_1)\vee (d_2\ominus d_1)]\).
In fact, suppose that \(M=\{[h_i,k_i]\mid c_2\le h_i\oplus c_1, d_2\le k_i\oplus d_1\}\), if \(\forall [h_i,k_i]\in M\), then \(c_2\le h_i\oplus c_1\) and \(d_2\le k_i\oplus d_1\), therefore \(c_2\ominus c_1\le h_i\le k_i\) and \(d_2\ominus d_1\le k_i\), hence \([c_2\ominus c_1,(c_2\ominus c_1)\vee (d_2\ominus d_1)]\le [h_i,k_i]\).
Therefore,
Secondly,
\([c_2\ominus c_1,(c_2\ominus c_1)\vee (d_2\ominus d_1)]\oplus _{SI} [c_1,d_1] =[(c_2\ominus c_1)\oplus c_1,((c_2\ominus c_1)\vee (d_2\ominus d_1))\oplus d_1] \ge [(c_2\ominus c_1)\oplus c_1,(d_2\ominus d_1)\oplus d_1]\ge [c_2,d_2]\), then
Finally, from (I) and (II),
For the first argument [a, b] of \(\gamma \), \([a,b]=\bigvee \{[e_i,f_i]\mid e_i\otimes a_1\le a_2, f_i\otimes b_1\le b_2, c_2\le h_i\oplus c_1, d_2\le k_i\oplus d_1, f_i\le 1-h_i, f_i\le 1-k_i\}\), so we can obtain that
Similarly, we have
From (III), (IV) and (V), it follows that
Moreover,
From Eq. (8),
Hence, this completes the proof from (VI) and (VII).
Appendix 3
Example 6
Let \(\otimes =\otimes _G\). Then,
Therefore, the result of implication \(\alpha \rightarrow _{SL^*}\beta \) could be computed and obtained as follows:
If \(a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([1,1],[0,0])\);
If \(a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([1-d_2,1-d_2],[0,d_2])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), \(\alpha \rightarrow _{SL^*}\beta =([1-c_2,1-c_2],[c_2,c_2])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([1-d_2,1-d_2],[c_2,d_2])\);
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([b_2,b_2],[0,0])\);
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([b_2,b_2],[0,d_2])\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([b_2,b_2],[c_2,c_2])\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([b_2,b_2],[c_2,d_2])\) ;
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,1],[0,0])\);
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,1-d_2],[0,d_2])\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,1-c_2],[c_2,c_2])\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,1-d_2],[c_2,d_2])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,b_2],[0,0])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,b_2],[0,d_2])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,b_2],[c_2,c_2])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([a_2,b_2],[c_2,d_2])\).
Example 7
Let \(\otimes =\otimes _{Lu}\). Then,
Thus, \(a=(1-a_1+a_2)\wedge (1-b_1+b_2)\wedge (1-c_2+c_1)\wedge (1-d_2+d_1)\wedge 1\),
Therefore,
Example 8
Let \(\otimes =\otimes _\pi \). Then,
Therefore, the result of implication \(\alpha \rightarrow _{SL^*}\beta \) could be computed and obtained as follows:
If \(a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([1,1],[0,0])\);
\(If a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{1-d_2}{1-d_1}, \frac{1-d_2}{1-d_1}],[0,\frac{d_2-d_1}{1-d_1}])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{1-c_2}{1-c_1}, \frac{1-c_2}{1-c_1}],[\frac{c_2-c_1}{c_2-c_1},\frac{c_2-c_1}{c_2-c_1}])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{1-c_2}{1-c_1} \wedge \frac{1-d_2}{1-d_1},\frac{1-c_2}{1-c_1}\wedge \frac{1-d_2}{1-d_1}], [\frac{c_2-c_1}{c_2-c_1},\frac{c_2-c_1}{c_2-c_1} \vee \frac{d_2-d_1}{1-d_1}])\);
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{b_2}{b_1},\frac{b_2}{b_1}],[0,0])\);
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{b_2}{b_1}\wedge \frac{1-d_2}{1-d_1},\frac{b_2}{b_1}\wedge \frac{1-d_2}{1-d_1}],[0,\frac{d_2-d_1}{1-d_1}])\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1},\frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1}], {[\frac{c_2-c_1}{1-c_1},\frac{c_2-c_1}{1-c_1}]})\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2> d_1\), \(\alpha \rightarrow _{SL^*}\beta =([\frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1}\wedge \frac{1-d_2}{1-d_1},\frac{b_2}{b_1} \wedge \frac{1-c_2}{1-c_1} \wedge \frac{1-d_2}{1-d_1}],[\frac{c_2-c_1}{1-c_1}, \frac{c_2-c_1}{1-c_1}\vee \frac{d_2-d_1}{1-d_1}])\);
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1},1],[0,0])\);
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{1-d_2}{1-d_1},\frac{1-d_2}{1-d_1}], {[0,\frac{d_2-d_1}{1-d_1}]})\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{1-c_2}{1-c_1},\frac{1-c_2}{1-c_1}],[\frac{c_2-c_1}{1-c_1}, \frac{c_2-c_1}{1-c_1}])\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{1-c_2}{1-c_1}\wedge \frac{1-d_2}{1-d_1},\frac{1-c_2}{1-c_1} \wedge \frac{1-d_2}{1-d_1}], [\frac{c_2-c_1}{1-c_1},\frac{c_2-c_1}{1-c_1}\vee \frac{d_2-d_1}{1-d_1}])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{b_2}{b_1},\frac{b_2}{b_1}],[0,0])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{b_2}{b_1}\wedge \frac{1-d_2}{1-d_1},\frac{b_2}{b_1} \wedge \frac{1-d_2}{1-d_1}], [0,\frac{d_2-d_1}{1-d_1}])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1},\frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1}], [\frac{c_2-c_1}{1-c_1},\frac{c_2-c_1}{1-c_1}])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([\frac{a_2}{a_1}\wedge \frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1}\wedge \frac{1-d_2}{1-d_1},\frac{b_2}{b_1}\wedge \frac{1-c_2}{1-c_1} \wedge \frac{1-d_2}{1-d_1}],[\frac{c_2-c_1}{1-c_1}, \frac{c_2-c_1}{1-c_1}\vee \frac{d_2-d_1}{1-d_1}])\).
Example 9
Let \(\otimes =\otimes _0\). Then,
Therefore, the result of implication \(\alpha \rightarrow _{SL^*}\beta \) could be computed and obtained as follows:
If \(a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([1,1],[0,0])\);
If \(a_1\le a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([d_1\vee (1-d_2),d_1 \vee (1-d_2)],[0,d_2\wedge (1-d_1)])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([(c_1\vee (1-c_2)),(c_1\vee (1-c_2))], [c_2\wedge (1-c_1),c_2\wedge (1-c_1)])\);
If \(a_1\le a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([(c_1\vee (1-c_2))\wedge (d_1 \vee (1-d_2)),(c_1\vee (1-c_2))\wedge (d_1\vee (1-d_2))], [c_2\wedge (1-c_1),(c_2\wedge (1-c_1))\vee (d_2\wedge (1-d_1))])\);
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([(1-b_1)\vee b_2,(1-b_1)\vee b_2],[0,0])\),
If \(a_1\le a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-b_1)\vee b_2)\wedge (d_1\vee (1-d_2)), ((1-b_1)\vee b_2)\wedge (d_1\vee (1-d_2))],[0,d_2\wedge (1-d_1)])\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), \(\alpha \rightarrow _{SL^*}\beta =([((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2)),((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2))],[c_2\wedge (1-c_1),c_2\wedge (1-c_1)])\);
If \(a_1\le a_2, b_1> b_2, c_2> c_1, d_2> d_1\), \(\alpha \rightarrow _{SL^*}\beta =([((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2)) \wedge (d_1\vee (1-d_2)),((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2))\wedge (d_1\vee (1-d_2))],[c_2 \wedge (1-c_1),(c_2\wedge (1-c_1)) \vee (d_2\wedge (1-d_1))])\);
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([(1-a_1)\vee a_2,1],[0,0])\);
If \(a_1> a_2, b_1\le b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge (d_1\vee (1-d_2)), d_1\vee (1-d_2)],[0,d_2\wedge (1-d_1)])\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge (c_1\vee (1-c_2)), c_1\vee (1-c_2)],[c_2\wedge (1-c_1),c_2\wedge (1-c_1)])\);
If \(a_1> a_2, b_1\le b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge (c_1\vee (1-c_2)) \wedge (d_1\vee (1-d_2)),(c_1\vee (1-c_2))\wedge (d_1 \vee (1-d_2))],[c_2\wedge (1-c_1), (c_2\wedge (1-c_1))\vee (d_2\wedge (1-d_1))])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge ((1-b_1)\vee b_2), (1-b_1)\vee b_2],[0,0])\);
If \(a_1> a_2, b_1> b_2, c_2\le c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge ((1-b_1)\vee b_2) \wedge (d_1\vee (1-d_2)),((1-b_1)\vee b_2)\wedge (d_1\vee (1-d_2))],[0,d_2\wedge (1-d_1)])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2\le d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge ((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2)),((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2))],[c_2\wedge (1-c_1),c_2\wedge (1-c_1)])\);
If \(a_1> a_2, b_1> b_2, c_2> c_1, d_2> d_1\), then \(\alpha \rightarrow _{SL^*}\beta =([((1-a_1)\vee a_2)\wedge ((1-b_1)\vee b_2) \wedge (c_1\vee (1-c_2))\wedge (d_1\vee (1-d_2)),((1-b_1)\vee b_2)\wedge (c_1\vee (1-c_2))\wedge (d_1\vee (1-d_2))], [c_2\wedge (1-c_1),(c_2\wedge (1-c_1))\vee (d_2\wedge (1-d_1))])\).
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Jin, J., Ye, M. & Pedrycz, W. Quintuple Implication Principle on interval-valued intuitionistic fuzzy sets. Soft Comput 24, 12091–12109 (2020). https://doi.org/10.1007/s00500-019-04649-1
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DOI: https://doi.org/10.1007/s00500-019-04649-1