Multigranulation Pythagorean fuzzy decisiontheoretic rough sets based on inclusion measure and their application in incomplete multisource information systems
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Abstract
Multigranulation rough sets (MGRSs) and decisiontheoretic rough sets (DTRSs) are two important and popular generalizations of classical rough sets. The combination of two generalized rough sets have been investigated by numerous researchers in different extensions of fuzzy settings such as intervalvalued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs), bipolarvalued fuzzy sets (BVFSs), etc. Pythagorean fuzzy (PF) set is another extension of fuzzy set, which is more capable in comparison to IFS handle vagueness in real world. However, few studies have focused on the combination of the two rough sets in PF settings. In this study, we combine the two generalized rough sets in PF settings. First, we introduce a type of PF subset (of subset of the given universe) of the PF Set (of the given universe). Then we establish two basic models of multigranulation PF DTRS (MGPFDTRS) of PF subset of the PF set based on PF inclusion measure within the framework of multigranulation PF approximation space. One model is based on a combination of PF relations (PFRs) and the construction of approximations with respect to the combined PFR. By combining PFRs through intersection and union, respectively, we construct two models. The other model is based on the construction of approximations from PFRs and a combination of the approximations. By using intersection and union to combine the approximations, respectively, we again get two models. As a result, we have total four models. Further for different constraints on parameters, we obtain three kinds of each model of the MGPFDTRSs. Then, their principal structure, basic properties and uncertainty measure methods are investigated as well. Second, we give a way to compute PF similarity degrees between two objects and also give a way to compute PF decisionmaking objects from incomplete multisource information systems (IMSISs). Then we design an algorithm for decisionmaking to IMSISs using MGPFDTRSs and their uncertainty measure methods. Finally, an example about the mutual funds investment is included to show the feasibility and potential of the theoretic results obtained.
Keywords
Pythagorean fuzzy set Pythagorean fuzzy inclusion measure Multigranulation Pythagorean fuzzy decisiontheoretic rough set Incomplete multisource information systemIntroduction
In light of Bayesian decision procedure [4], decision theoretic rough set (DTRS) model was proposed by Yao and Wong [35] to analyze the noisy data by considering the tolerance of classification error. Since then the DTRS model has found its applications in various theoretical and practical fields, and it has produced many god results [2, 15, 16, 17, 20, 25].
However, the DTRS model cannot deal with numerical data directly. To overcome this disadvantage, researchers used tolerance relations [21], similarity measures [23], domains relations [3], covering [33], inclusion measures [38], fuzzy relations [32, 41, 42], fuzzy preference relations [26], intervalvalued fuzzy preference relations [29], intuitionistic fuzzy relations [19, 40], intuitionistic fuzzy inclusion measure [13, 13], bipolarvalued fuzzy relations [24, 28] in place of equivalence relations.
To handle all types of real data, Yager proposed the concepts of Pythagorean fuzzy set (PFS) [34], which are more powerful than intuitionistic fuzzy sets (IFSs) [1] for dealing with the uncertain information in decisionmaking procedures. For example, if a decision maker gives the membership degree and nonmembership degree as 0.8 and 0.6, respectively, then it is only valid for PFS. Fortunately, PFSs generalize the concept of IFSs and the corresponding operational laws, which have been successfully applied to some complex practical decisionmaking situations, e.g., roadbuilding projects [11], selection of the optimal production strategy [12] and group decisionmaking problems [8]. Besides, Zhang and Ren [37] investigated Pythagorean fuzzy multigranulation rough set over two universes and its applications in merger and acquisition. Liang et al. [18] gave a method of threeway decisions using ideal TOPSIS solutions on Pythagorean fuzzy informations. Mandal and Ranadive [25] studied decisiontheoretic rough sets under Pythagorean fuzzy information.
 1.
our model can deal with both intuitionistic fuzzy and Pythagorean fuzzy information instead of only intuitionistic fuzzy information;
 2.
our model can deal with complete and incomplete multisource information systems instead of only complete multisource information systems;
 3.
instead of assuming a fuzzy decisionmaking object as many researchers do, we give a method to find it.
Preliminaries
In this section, we present some basic concepts and terminology used throughout the paper.
Definition 1
Throughout this paper by PFS(U) we mean the set of all PFSs defined on U.
Definition 2
 1.
\(A^{c} = \{\left<x, \nu _{A}(x), \mu _{A}(x)\right> \mid x \in U\}\);
 2.
\( A \subseteq B \) if \(\forall x \in U\), \(\mu _{A}(x) \le \mu _{B}(x)\) and \(\nu _{A}(x) \ge \nu _{B}(x)\);
 3.
\( A = B\) iff \(\forall x \in U\), \(\mu _{A}(x) = \mu _{B}(x)\) and \(\nu _{A}(x) = \nu _{B}(x)\);
 4.
\(\Phi _{A}=\{\left<x, 1,0\right>\mid x \in U\}\);
 5.
\(\emptyset _{A}=\{\left<x,0,1\right>\mid x \in U\}\);
 6.
\(A \cap B = \{\left<x, \mu _{A}(x) \wedge \mu _{B}(x), \nu _{A}(x) \vee \nu _{B}(x)\right> \mid x \in U\}\);
 7.
\(A \cup B = \{\left<x, \mu _{A}(x) \vee \mu _{B}(x), \nu _{A}(x) \wedge \nu _{B}(x)\right> \mid x \in U\}\).
Thus it is clear that if \(X \subseteq U\), then \(A(X), A(X^{c}) \subseteq A \).
Definition 3
The Pythagorean fuzzy relation on U
\(R(U \times U)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\) 

\(x_{1}\)  \(\left<1,0\right>\)  \(\left<0.5,0.7\right>\)  \(\left<0.6,0.7\right>\)  \(\left<0.4,0.3\right>\)  \(\left<0.5,0.3\right>\) 
\(x_{2}\)  \(\left<0.5,0.7\right>\)  \(\left<1,0\right>\)  \(\left<0.5,0.4\right>\)  \(\left<0.5,0.6\right>\)  \(\left<0.6,0.4\right>\) 
\(x_{3}\)  \(\left<0.6,0.7\right>\)  \(\left<0.5,0.4\right>\)  \(\left<1,0\right>\)  \(\left<0.7,0.6\right>\)  \(\left<0.6,0.5\right>\) 
\(x_{4}\)  \(\left<0.4,0.3\right>\)  \(\left<0.5,0.6\right>\)  \(\left<0.7,0.6\right>\)  \(\left<1,0\right>\)  \(\left<0.4,0.3\right>\) 
\(x_{5}\)  \(\left<0.5,0.3\right>\)  \(\left<0.6,0.4\right>\)  \(\left<0.6,0.5\right>\)  \(\left<0.4,0.3\right>\)  \(\left<1,0\right>\) 
The Pythagorean fuzzy inclusion measure is also called the Pythagorean fuzzy subsethood measure, which indicates the degree to which one PFS is contained in another PFS. Peng et al. [31] provided a simple definition of the Pythagorean fuzzy inclusion measure as follows:
Definition 4
 1.
\(0 \le I(A(U),B(U)) \le 1\);
 2.
\(I(A,B) = 1\) iff \(A \subseteq B\);
 3.
\(I(A,B) = 0\) iff \(A = \Phi _{A}\), \(B = \emptyset _{B}\);
 4.
If \(A \subseteq B \subseteq C\), then \(I(C,A)\le I(B,A)\) and \(I(C, A)\le I(C,B)\).
We provide an inclusion measure on \(\mathrm{PFS}(U)\) on the basis of the Theorems 3.7(10) and 3.22 presented by Peng et al. [31].
Definition 5
The Pythagorean fuzzy relation on \(U \times X\)
\(R(U \times X)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\) 

\(x_{1}\)  \(\left<0,1\right>\)  \(\left<0,1\right>\)  \(\left<0.6,0.7\right>\)  \(\left<0,1\right>\)  \(\left<0.5,0.3\right>\) 
\(x_{2}\)  \(\left<0,1\right>\)  \(\left<0,1\right>\)  \(\left<0.5,0.4\right>\)  \(\left<0,1\right>\)  \(\left<0.6,0.4\right>\) 
\(x_{3}\)  \(\left<0,1\right>\)  \(\left<0,1\right>\)  \(\left<1,0\right>\)  \(\left<0,1\right>\)  \(\left<0.6,0.5\right>\) 
\(x_{4}\)  \(\left<0,1\right>\)  \(\left<0,1\right>\)  \(\left<0.7,0.6\right>\)  \(\left<0,1\right>\)  \(\left<0.4,0.3\right>\) 
\(x_{5}\)  \(\left<0,1\right>\)  \(\left<0,1\right>\)  \(\left<0.6,0.5\right>\)  \(\left<0,1\right>\)  \(\left<1,0\right>\) 
For example, two PFSs \(A=\{\left<x_{1}, 0.9, 0.3\right>\), \(\left<x_{2}, 0.4, 0.7\right>\), \(\left<x_{3}, 0.8, 0.4\right>\), \(\left<x_{4}, 0.7, 0.2\right>\}\) and \(B=\{\left<x_{1}, 0.7, 0.6\right>\), \(\left<x_{2}, 0.9, 0.2\right>\), \(\left<x_{3}, 0.7, 0.5\right>\), \(\left<x_{4}, 0.8, 0.2\right>\}\). Then \(I(A,B)= \frac{0.49+0.16+0.49+0.49+0.09+0.49+0.16+0.04}{0.81+0.16+0.64+0.49+0.36+0.49+0.25+0.04} = \frac{2.41}{3.24}=0.74382716\)
MGPFDTRSs based on inclusion measure
In this section first we propose and study the models of inclusion measurebased MGPFDTRSs, within the framework of multigranulation Pythagorean fuzzy approximation space.
Definition 6
Let U be a finite universe. For any \(X\subseteq U\) and \(R_{k}(U \times X)(1\le k \le m)\) be m Pythagorean fuzzy relations on \(U \times X\). Then, we call \((U,X,R_{k}(U \times X)(1\le k \le m))\) a multigranulation Pythagorean fuzzy approximation space on U.
Definition 7
Now in the following we study four types PFDTRSs based on the inclusion measure, within the framework of multigranulation Pythagorean fuzzy approximation space, i.e., four types of MGPFDTRSs.
TypeI MGPFDTRSs
Definition 8
Remark 1
Remark 2
 1.For any \(X \subseteq U\), we obtain$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)} (A(X)) \subseteq \overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)). \end{aligned}$$
 2.
\(\underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times U)}(U)=U \overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times U)}(U)\).
 3.For any \(X \subseteq U\) and \(0\le \beta _{1}\le \beta _{2}\le \alpha _{1}\le \alpha _{2}\le 1\), we obtainand$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha _{2}}_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \underline{\mathrm{Apr}}^{\alpha _{1}}_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)). \end{aligned}$$$$\begin{aligned} \overline{\mathrm{Apr}}^{\beta _{2}}_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \overline{\mathrm{Apr}}^{\beta _{1}}_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)). \end{aligned}$$
 4.For any \(X,Y \subseteq U\) with \(A(X) \subseteq A(Y)\), we obtainand$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(Y)) \end{aligned}$$$$\begin{aligned} \overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(Y)). \end{aligned}$$
Remark 3
 1.The TypeI \(\alpha \)MGPFDTRS (with \(0.5 < \alpha \le 1\)) of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measure based TypeI multigranulation Pythagorean fuzzy \(\alpha \)lower and \(\alpha \)upper approximations as$$\begin{aligned}&\underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cap _{k=1}^{m}R_{k}(U \times X)},A(X))\ge \alpha \},\\&\overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cap _{k=1}^{m}R_{k}(U \times X)},A(X))> 1 \alpha \}. \end{aligned}$$
 2.The TypeI 0.5MGPFDTRS of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measure based TypeI multigranulation Pythagorean fuzzy 0.5lower and 0.5upper approximations as$$\begin{aligned}&\underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cap _{k=1}^{m}R_{k}(U \times X)},A(X)) >0.5 \},\\&\overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cap _{k=1}^{m}R_{k}(U \times X)},A(X))\ge 0.5\}. \end{aligned}$$
TypeII MGPFDTRSs
Definition 9
Remark 4
Remark 5
 1.For any \(X \subseteq U\), we obtain$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)). \end{aligned}$$
 2.
\(\underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times U)}(U)=U= \overline{\mathrm{Ap}r}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times U)}(U)\).
 3.For any \(X \subseteq U\) and \(0\le \beta _{1}\le \beta _{2}\le \alpha _{1}\le \alpha _{2}\le 1\), we obtainand$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha _{2}}_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \underline{\mathrm{Apr}}^{\alpha _{1}}_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)) \end{aligned}$$$$\begin{aligned} \overline{\mathrm{Apr}}^{\beta _{2}}_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X))\subseteq \overline{\mathrm{Apr}}^{\beta _{1}}_{\cup _{k=1}^{m}R_{k}}(A(X)). \end{aligned}$$
 4.For any \(X, Y \subseteq U\) with \(A(X) \subseteq A(Y)\), we obtainand$$\begin{aligned} \underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(Y)) \end{aligned}$$$$\begin{aligned} \overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X)) \subseteq \overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(Y)). \end{aligned}$$
Remark 6
 1.The TypeII \(\alpha \)MGPFDTRS (with \(0.5 < \alpha \le 1\)) of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measurebased TypeII multigranulation Pythagorean fuzzy \(\alpha \)lower and \(\alpha \)upper approximations as$$\begin{aligned}&\underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\left\{ x \in U: I([x]_{\cup _{k=1}^{m}R_{k}(U \times X)},A(X))\ge \alpha \right\} ,\\&\overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\left\{ x \in U: I([x]_{\cup _{k=1}^{m}R_{k}(U \times X)},A(X))> 1 \alpha \right\} . \end{aligned}$$
 2.The TypeII 0.5MGPFDTRS of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measurebased TypeII multigranulation Pythagorean fuzzy 0.5lower and 0.5upper approximations as$$\begin{aligned}&\underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cup _{k=1}^{m}R_{k}(U \times X)},A(X)) >0.5 \},\\&\overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: I([x]_{\cup _{k=1}^{m}R_{k}(U \times X)},A(X))\ge 0.5\}. \end{aligned}$$
TypeIII MGPFDTRSs
Definition 10
Remark 7
Remark 8
 1.For any \(X\subseteq U\), we obtain$$\begin{aligned} \cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X)) \subseteq \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X)). \end{aligned}$$
 2.
\(\cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times U)}(U)=U \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times U)}(U)\).
 3.For any \(X\subseteq U\) and \(0\le \beta _{1}\le \beta _{2}\le \alpha _{1}\le \alpha _{2}\le 1\), we obtainand$$\begin{aligned} \cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha _{2}}_{R_{k}}(A)\subseteq \cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha _{1}}_{R_{k}(U \times X)}(A(X)) \end{aligned}$$$$\begin{aligned} \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta _{2}}_{R_{k}(U \times X)}(A(X)) \subseteq \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta _{1}}_{R_{k}(U \times X)}(A(X)). \end{aligned}$$
 4.For any \(X,Y \subseteq U\) with \(A(X) \subseteq B(X)\), we obtainand$$\begin{aligned} \cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X)) \subseteq \cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}\,(A(Y)) \end{aligned}$$$$\begin{aligned} \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X)) \subseteq \cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(Y)). \end{aligned}$$
Remark 9
 1.the TypeIII \(\alpha \)MGPFDTRS (with 0.5 \( < \alpha \le 1\)) of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measurebased TypeIII multigranulation Pythagorean fuzzy \(\alpha \)lower and \(\alpha \)upper approximations as$$\begin{aligned}&\cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cap _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X))\ge \alpha \},\\&\cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cup _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X)) > 1\alpha \}. \end{aligned}$$
 2.the TypeIII 0.5MGPFDTRS of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measurebased TypeIII multigranulation Pythagorean fuzzy 0.5lower and 0.5upper approximations as$$\begin{aligned}&\cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X))\\&\quad =\left\{ x \in U: \cap _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X)) >0.5 \right\} ,\\&\cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X))\\&\quad =\left\{ x \in U: \cup _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X))\ge 0.5\right\} . \end{aligned}$$
TypeIV MGPFDTRSs
Definition 11
Remark 10
 1.
\(\cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times U)}(U)=U \cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times U)}(U)\).
 2.For any \(X\subseteq U\) and \(0\le \beta _{1}\le \beta _{2}\le \alpha _{1}\le \alpha _{2}\le 1\), we obtainand$$\begin{aligned} \cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha _{2}}_{R_{k}(U \times X)}(A(X)) \subseteq \cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha _{1}}_{R_{k}(U \times X)}(A(X)) \end{aligned}$$$$\begin{aligned} \cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta _{2}}_{R_{k}(U \times X)}(A(X))\subseteq \cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta _{1}}_{R_{k}(U \times X)}(A(X)). \end{aligned}$$
 3.For any \(X,Y \subseteq U\) with \(A(X) \subseteq A(Y)\), we obtainand$$\begin{aligned} \cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X)) \subseteq \cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}\,(A(Y)) \end{aligned}$$$$\begin{aligned} \cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X)) \subseteq \cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(Y)). \end{aligned}$$
Remark 11
 1.the TypeIV \(\alpha \)MGPFDTRS (with \(0.5 < \alpha \le 1\)) of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X \subseteq U\) in terms of inclusion measurebased TypeIV multigranulation Pythagorean fuzzy \(\alpha \)lower and \(\alpha \)upper approximations as$$\begin{aligned}&\cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cup _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X))\ge \alpha \},\\&\cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cap _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X)) > 1\alpha \}. \end{aligned}$$
 2.the TypeIV 0.5MGPFDTRS of A(X) w.r.t. \((U,X,R_{k}(U \times X)(1\le k \le m))\) is defined for any \(X\subseteq U\) in terms of inclusion measurebased TypeIV multigranulation Pythagorean fuzzy 0.5lower and 0.5upper approximations as$$\begin{aligned}&\cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cup _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X)) >0.5 \},\\&\cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X))\\&\quad =\{x \in U: \cap _{k=1}^{m}I([x]_{R_{k}(U \times X)},A(X))\ge 0.5\}. \end{aligned}$$
Uncertainty measures
In this section, several measures are utilized to calculate the uncertainty of these models which is discussed in previous section. The uncertainty of knowledge is caused by the boundary regions, in the view point of approximations. The larger the boundary area is, the more the uncertainty. The accuracy, roughness and approximation quality are studied in the next.
Definition 12
Definition 13
Definition 14
Decisionmaking to incomplete multisource information systems using MGPFDTRSs
In this section, based on the MGPFDTRSs and their uncertainty measures established in “MGPFDTRSs based on inclusion measure” and “Uncertainty measures”, we will construct a new method and approach to decisionmaking with incomplete multisource information systems. Also, we will present the decisionmaking algorithm and the general steps for established method in detail.
Incomplete multisource information systems and the similarity degrees
Definition 15
 1.
U is a finite nonempty set of objects, called the universe;
 2.
\(\mathrm{AT}_{l}\) is a nonempty finite set of attributes of each subsystem;
 3.
\(\{V_{a}\}\) is the domain of the attribute \(a \in \mathrm{AT}_{l}\); and
 4.
\(f_{l}: U \times \mathrm{AT}_{l} \mapsto \{(V_{a})_{a \in \mathrm{AT}_{l}}\}\) such that for all \(x \in U\) and \(a \in \mathrm{AT}_{l}\), \(f(x, a) \in V_{a}\).
Definition 16
An incomplete multisource information system (IMSIS) indicates the precise attribute values \(V_{a}\) for some objects are unknown. In this paper, the IMSIS is still denoted without confusion by \(\mathrm{IMSIS}=\{\mathrm{IS}_{l} \mid \mathrm{IS}_{l}=(U, \mathrm{AT}_{l}, V, f_{l}\}\). Here \(V=\{(V_{a})_{a \in \mathrm{AT}_{l}}\} \cup \{*\}\), the special symbol “\(*\)” is used to indicate the unknown value. For instance, if \(f(x, a)=*\), the value of object x is unknown on the attribute a.
Liu et al. [23] handle the incomplete single source information system and compute the similarity degree between two objects. Their similarity degree between two objects is fuzzy set. Here, we handle the incomplete multisource information system and compute the similarity degree between two objects. Our similarity degree between two objects is PFS.
 1.
Consideration of \((x, a_{i})\ne *\) and \((y, a_{i})\ne *\), \((x, a_{i})\) and \((y, a_{i})\) are equality iff \((x, a_{i})=(y, a_{i})\);
 2.
Consideration of \((x, a_{i})\ne *\) and \((y, a_{i})\ne *\), \((x, a_{i})\) and \((y, a_{i})\) are not the same if \((x, a_{i}) \ne (y, a_{i})\);
 3.
Consideration of \((x, a_{i}) = *\) or \((y, a_{i}) = *\), because of the unknown value “\(*\)” is treated as “do not care” conditions, it has the probability of \(\frac{1}{\left V_{a_{i}}\right }\) to equal to one certain value of \(V_{a_{i}}\) (\(V_{a_{i}}\) is a domain of the attribute \(a_{i}\), \(\left V_{a_{i}}\right \) denotes the cardinality of \(a_{i}\)).
 4.
Consideration of \((x, a_{i})=(y, a_{i})=*\), both of \(a_{i}(x)\) and \(a_{i}(y)\) have the probability of \(\frac{1}{\left V_{a_{i}}\right }\) to equal to one certain value of \(V_{a_{i}}\), so the joint probability of \((x, a_{i})=(y, a_{i})\) is \(\frac{1}{\left V_{a_{i}}\right ^{2}}\).
Example 1
Let us consider an evaluation problem of a car depicted by an IMSIS presented in Table 3. Suppose that \(U=\{x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\), \(x_{5}\), \(x_{6}\}\) is a set of six cars. Every car in each subinformation (source) system, denoted by \(\mathrm{EC}_{1}\) and \(\mathrm{EC}_{2}\), is described by two attributes. They are \(c_{1}=\text {Price}\), \(c_{2}=\text {Mileage}\), \(c_{3}=\text {Size}\), \(c_{4}=\text {Maxspeed}\), respectively. The domains of the attributes are as follows: \(V_{c_{1}}=\{\text {High}, \text {Low}\}\), \(V_{c_{2}}=\{\text {High}, \text {Low}\}\), \(V_{c_{3}}=\{\text {Full}, \text {Medium}, \text {Compact}\}\), \(V_{c_{4}}=\{\text {High}, \text {Low}\}\).
A car incomplete multisource information system
U  \(\mathrm{EC}_{1}\)  \(\mathrm{EC}_{2}\)  

\(c_{1}\)  \(c_{3}\)  \(c_{2}\)  \(c_{4}\)  
\(x_{1}\)  High  Full  Low  Low 
\(x_{2}\)  Low  Medium  \(*\)  Low 
\(x_{3}\)  \(*\)  Compact  \(*\)  Low 
\(x_{4}\)  High  Full  \(*\)  High 
\(x_{5}\)  \(*\)  Full  \(*\)  High 
\(x_{6}\)  Low  Compact  High  \(*\) 
The computing results of \(R_{\mathrm{EC}_{1}}(U)\)
\(R_{\mathrm{EC}_{1}}(U \times U)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\) 

\(x_{1}\)  \(\left<1,0\right>\)  \(\left<0,1\right>\)  \(\left<0.2500,0.9330\right>\)  \(\left<1,0\right>\)  \(\left<0.7500,0.4303\right>\)  \(\left<0,1\right>\) 
\(x_{2}\)  \(\left<1,0\right>\)  \(\left<0.2500,0.6330\right>\)  \(\left<0,1\right>\)  \(\left<0.2500,0.9330\right>\)  \(\left<0.5000,0.5000\right>\)  
\(x_{3}\)  \(\left<1,0\right>\)  \(\left<0.2500,0.9330\right>\)  \(\left<0.1250,0.9841\right>\)  \(\left<0.2500,0.9330\right>\)  
\(x_{4}\)  \(\left<1,0\right>\)  \(\left<0.7500,0.4330\right>\)  \(\left<0,1\right>\)  
\(x_{5}\)  \(\left<1,0\right>\)  \(\left<0.2500,0.9330\right>\)  
\(x_{6}\)  \(\left<1,0\right>\) 
In the same way, we also compute \(\mu _{R_{\mathrm{EC}_{2}}(U)}\) \((x_{i},x_{j})\) and \(\nu _{R_{\mathrm{EC}_{}}(U)}(x_{i},x_{j})\) for all \(x_{i}\), \(x_{j} \in U\) (i, \(j = 1, 2, \ldots , 5)\) but here it is not necessary.
Based on the basic principle of MGPFDTRSs, a Pythagorean fuzzy decisionmaking object is approximated over the multigranulation Pythagorean fuzzy approximation space. So, we give the approach for compute the Pythagorean fuzzy decisionmaking object from the IMSIS. Then we give the following algorithm for compute the degrees of membership and nonmembership of any alternative with respect to a Pythagorean fuzzy decisionmaking object from the IMSIS:
Algorithm 1
 Step 1:

First we consider the reasonable membership degree of the domains of the all attributes.
 Step 2:

If the membership degree of the domain of the attribute \(a_{i}=\alpha _{i}\), i.e. \((x, a_{i})=\alpha _{i}\), then we write \(\mu _{A}(x, a_{i})=\alpha _{i}\) and compute \(\nu _{A}(x, a_{i})=\sqrt{1\alpha _{i}^{2}}\). If \((x, a_{i})=*\), then we consider \(\mu _{A}(x, a_{i})=\frac{1}{\left V_{a_{i}}\right }\) and \(\nu _{A}(x\), \(a_{i})=\sqrt{1\frac{1}{\left V_{a_{i}}\right ^{2}}}\).
 Step 3:
 The degrees of membership and nonmembership of any \(x \in U\) is calculated as follows:$$\begin{aligned} \mu _{A}(x)&=\sum _{i=1}^{t} \frac{\mu _{A(U)}(x,a_{i})}{t}, \end{aligned}$$(31)$$\begin{aligned} \nu _{A}(x)&=\sum _{i=1}^{t} \frac{\nu _{A(U)}(x,a_{i})}{t}. \end{aligned}$$(32)
For clearance we have the following example:
Example 2
 Step 1:

Let us we consider the reasonable membership degree of the domains of the all attributes, which is shown in Fig. 2.
 Step 2:

From Table 3, we get \((x_{1},c_{1})=\text {High}=1\), i.e. \(\mu _{A}(x_{1},c_{1})=1\), then \(\nu _{A}(x_{1},c_{1})=0\). Similarly, \(\mu _{A}(x_{1},c_{2})=0\), \(\nu _{A}(x_{1},c_{2})=1\), \(\mu _{A}(x_{1},c_{3})=1\), \(\nu _{A}(x_{1},c_{3})=0\), \(\mu _{A}(x_{1},c_{4})=0\), \(\nu _{A}(x_{1},c_{4})=1\).
 Step 3:
 Using Eqs. (31) and (32), we have \(\mu _{A}(x_{1})=\frac{1+0+1+0}{4}=0.5\) and \(\nu _{A}(x_{1})=\frac{0+1+0+1}{4}=0.5\). Similarly, we also find \(\mu _{A}(x_{i})\) and \(\nu _{A}(x_{i})\) \(i=\{2,3, \ldots , 6\}\), which is represented in Eq. (32).$$\begin{aligned} A=&\frac{\left<0.5,0.5\right>}{x_{1}}+\frac{\left<0.25,0.93\right>}{x_{2}} \nonumber \\&+\frac{\left<0.44,0.89\right>}{x_{3}} +\frac{\left<0.88,0.27\right>}{x_{4}} \nonumber \\&+\frac{\left<0.75,0.43\right>}{x_{5}}+\frac{\left<0.65,0.63\right>}{x_{6}}. \end{aligned}$$(33)
An algorithm
 Step 1:

Suppose that a decision making problem the IMSIS is \(\mathrm{MSIS}=\{\mathrm{IS}_{l} \mid \mathrm{IS}_{l}=(U, \mathrm{AT}_{l}, V=\{(V_{a})_{a \in \mathrm{AT}_{l}}\}\cup \{*\}, f_{l}\}\). Let us assume that \(X_{1}\), \(X_{2}, \ldots , X_{r}\) be the subsets of U, where the elements of \(X_{i}\) \((i=1, 2, \ldots , r)\) are randomly selected and not repeat any other elements in \(X_{i}\) \((i=1, 2, \ldots \), r). To find the best \(X_{i}\) \((i=1, 2, \ldots , r)\).
 Step 2:

Computing \(R_{k}(1 \le k \le l)(U)\) according to Eq. (30).
 Step 3:

Constructing the Pythagorean fuzzy decisionmaking object A according to Algorithm 1.
 Step 4:

Choose \(\alpha \) and \(\beta \).
 Step 5:

Computing the inclusion measure based TypeI multigranulation Pythagorean fuzzy \(\alpha \)lower approximation
\(\underline{\mathrm{Apr}}^{\alpha }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X_{i}))\)
and \(\beta \)upper approximation
\(\overline{\mathrm{Apr}}^{\beta }_{\cap _{k=1}^{m}R_{k}(U \times X)}(A(X_{i}))\)
for each \(X_{i} \subseteq U\), respectively.
 Step 6:

Computing the inclusion measurebased TypeII multigranulation Pythagorean fuzzy \(\alpha \)lower approximation
\(\underline{\mathrm{Apr}}^{\alpha }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X_{i}))\)
and \(\beta \)upper approximation
\(\overline{\mathrm{Apr}}^{\beta }_{\cup _{k=1}^{m}R_{k}(U \times X)}(A(X_{i}))\)
for each \(X_{i} \subseteq U\), respectively.
 Step 7:

Computing the inclusion measurebased TypeIII multigranulation Pythagorean fuzzy \(\alpha \)lower approximation
\(\cap _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X_{i}))\)
and \(\beta \)upper approximation
\(\cup _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X_{i}))\)
for each \(X_{i} \subseteq U\), respectively.
 Step 8:

Computing the inclusion measure based TypeIII multigranulation Pythagorean fuzzy \(\alpha \)lower approximation
\(\cup _{k=1}^{m}\underline{\mathrm{Apr}}^{\alpha }_{R_{k}(U \times X)}(A(X_{i}))\)
and \(\beta \)upper approximation
\(\cap _{k=1}^{m}\overline{\mathrm{Apr}}^{\beta }_{R_{k}(U \times X)}(A(X_{i}))\)
for each \(X_{i} \subseteq U\), respectively.
 Step 9:

Computing \(\rho _{\mathrm{I}}\), \(\rho _{\mathrm{II}}\), \(\rho _{\mathrm{III}}\) and \(\rho _{\mathrm{IV}}\) using Eqs. (14)–( 17).
 Step 10:

Computing \(\sigma _{\mathrm{I}}\), \(\sigma _{\mathrm{II}}\), \(\sigma _{\mathrm{III}}\) and \(\sigma _{\mathrm{IV}}\) using Eqs. (18)–( 21).
 Step 11:

Computing \(\omega _{\mathrm{I}}\), \(\omega _{\mathrm{II}}\), \(\omega _{\mathrm{III}}\) and \(\omega _{\mathrm{IV}}\) using Eqs. (22)–( 25).
 Step 12:

Obtain the best \(X_{i}\) according to the higher accuracy, approximation degree and approximation quality for each \(X_{i}\).
An illustrative example
In this subsection, we apply the proposed algorithm to a real decision making. This example is about quick decision making based on a real investment context, under the MGPFDTRSs and their uncertainty measures models, where the information comes from multiple and incomplete.
Problem description
The various types of mutual funds (MFs) of different companies listed in the Growth Enterprise Market board of the India Stock Exchange are a popular investment source to an investor as a longterm investment. However, the sufficient knowledge about the various types of MFs of different companies is always not possible for every investor. Our proposed models are effective for those investors. Suppose an investor plans to invest his/her money in MFs of different companies, with the aim of high returns, while he/she has no sufficient knowledge about all MFs, then He/she chooses initially ten MFs according to the past performances, while he/she invests his/her money intpp the best five MFs out of these ten MFs. For making reasonable five MFs out of ten MFs, we have the following decision analysis.
Decision analysis
We use the algorithm in “An algorithm” of decision analysis based on MGPFDTRSs and their uncertainty measure, for decision making.
Incomplete multisource information system for ten MFs
U  \(\mathrm{EC}_{1}\)  \(\mathrm{EC}_{2}\)  \(\mathrm{EC}_{3}\)  

\(c_{1}\)  \(c_{2}\)  \(c_{6}\)  \(c_{7}\)  \(c_{3}\)  \(c_{5}\)  \(c_{8}\)  \(c_{4}\)  \(c_{9}\)  \(c_{10}\)  
\(x_{1}\)  High  Average  Fine  Moderate  Moderate  Low  Fine  Fine  Moderately high  Fine 
\(x_{2}\)  High  \(*\)  Fine  Moderately high  Moderate  \(*\)  Fine  Good  High  \(*\) 
\(x_{3}\)  Low  \(*\)  Fine  \(*\)  \(*\)  Moderate  \(*\)  Good  Moderately low  Good 
\(x_{4}\)  \(*\)  Low  Poor  \(*\)  \(*\)  Moderately low  \(*\)  Fine  \(*\)  Fine 
\(x_{5}\)  High  Average  Good  Low  \(*\)  High  Good  \(*\)  High  \(*\) 
\(x_{6}\)  \(*\)  Low  Fine  \(*\)  Moderately low  \(*\)  \(*\)  Fine  Moderate  \(*\) 
\(x_{7}\)  \(*\)  \(*\)  Good  High  \(*\)  \(*\)  Fine  \(*\)  Moderately high  Fine 
\(x_{8}\)  Low  \(*\)  Fine  Moderately high  Moderate  \(*\)  Good  Fine  \(*\)  \(*\) 
\(x_{9}\)  \(*\)  Average  Good  \(*\)  \(*\)  Low  Poor  \(*\)  Low  \(*\) 
\(x_{10}\)  Low  Low  Fine  \(*\)  \(*\)  Moderate  \(*\)  Fine  Moderate  Fine 
The computing result of \(R_{1}(U)\) from \(\mathrm{EC}_{1}\)
\(R_{1}(U)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\)  \(x_{7}\)  \(x_{8}\)  \(x_{9}\)  \(x_{10}\) 

\(x_{1}\)  \(<1\),  \(<0.6250\),  \(<0.4250\),  \(<0.1750\),  \(<0.5000\),  \(<0.4250\),  \(<0.2500\),  \(<0.3750\),  \(<0.4250\),  \(<0.3000\), 
\(0>\)  \(0.4665>\)  \(0.7115>\)  \(0.9615>\)  \(0.5000>\)  \(0.7115>\)  \(0.9330>\)  \(0.7165>\)  \(0.7115>\)  \(0.7449>\)  
\(x_{2}\)  \(<1\),  \(<0.3625\),  \(<0.3000\),  \(<0.3750\),  \(<0.5500\),  \(<0.1875\),  \(<0.5625\),  \(<0.3000\),  \(<0.4250\),  
\(0>\)  \(0.7370>\)  \(0.9280>\)  \(0.7165>\)  \(0.6780>\)  \(0.9586>\)  \(0.4921>\)  \(0.9280>\)  \(0.7115>\)  
\(x_{3}\)  \(<1\),  \(<0.2600\),  \(<0.1750\),  \(<0.5100\),  \(<0.2375\),  \(<0.6125\),  \(<0.2600\),  \(<0.635\),  
\(0>\)  \(0.9328>\)  \(0.9615>\)  \(0.6828>\)  \(0.9535>\)  \(0.4870>\)  \(0.9328>\)  \(0.4663>\)  
\(x_{4}\)  \(<1\),  \(<0.1750\),  \(<0.3225\),  \(<0.2375\),  \(<0.3000\),  \(<0.0725\),  \(<0.385\),  
\(0>\)  \(0.9615>\)  \(0.7419>\)  \(0.9535>\)  \(0.9280>\)  \(0.9919>\)  \(0.7163>\)  
\(x_{5}\)  \(<1\),  \(<0.1750\),  \(<0.5000\),  \(<0.1250\),  \(<0.6750\),  \(<0.0500\),  
\(0>\)  \(0.9615>\)  \(0.6380>\)  \(0.9665>\)  \(0.4615>\)  \(0.9949>\)  
\(x_{6}\)  \(<1\),  \(<0.2375\),  \(<0.5500\),  \(<0.0725\),  \(<0.6350\),  
\(0>\)  \(0.9535>\)  \(0.6780>\)  \(0.9919>\)  \(0.4663>\)  
\(x_{7}\)  \(<1\),  \(<0.1875\),  \(<0.4875\),  \(<0.3000\),  
\(0>\)  \(0.9586>\)  \(0.7035>\)  \(0.9280>\)  
\(x_{8}\)  \(<1\),  \(<0.3000\),  \(<0.6750\),  
\(0>\)  \(0.9280>\)  \(0.4615>\)  
\(x_{9}\)  \(<1\),  \(<0.1350\),  
\(0>\)  \(0.9663>\)  
\(x_{10}\)  \(<1\),  
\(0>\) 
The computing result of \(R_{2}(U)\) from \(\mathrm{EC}_{2}\)
\(R_{2}(U)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\)  \(x_{7}\)  \(x_{8}\)  \(x_{9}\)  \(x_{10}\) 

\(x_{1}\)  \(<1\),  \(<0.7333\),  \(<0.2778\),  \(<0.2778\),  \(<0.1667\),  \(<0.1778\),  \(<0.5667\),  \(<0.4000\),  \(<0.5000\),  \(<0.2778\), 
\(0>\)  \(0.3266>\)  \(0.9363>\)  \(0.9363>\)  \(0.9553>\)  \(0.9742>\)  \(0.6153>\)  \(0.6599>\)  \(0.6220>\)  \(0.9363>\)  
\(x_{2}\)  \(<1\),  \(<0.3444\),  \(<0.3444\),  \(<0.2333\),  \(<0.1244\),  \(<0.5133\),  \(<0.3467\),  \(<0.2333\),  \(<0.3444\),  
\(0>\)  \(0.9295>\)  \(0.9295>\)  \(0.9486>\)  \(0.9807>\)  \(0.6217>\)  \(0.6664>\)  \(0.9486>\)  \(0.9295>\)  
\(x_{3}\)  \(<1\),  \(<0.1204\),  \(<0.1944\),  \(<0.2704\),  \(<0.2611\),  \(<0.3444\),  \(<0.1944\),  \(<0.4537\),  
\(0>\)  \(0.9874>\)  \(0.9704>\)  \(0.9465>\)  \(0.9636>\)  \(0.9295>\)  \(0.9704>\)  \(0.6540>\)  
\(x_{4}\)  \(<1\),  \(<0.1204\),  \(<0.2704\),  \(<0.2611\),  \(<0.3444\),  \(<0.1944\),  \(<0.1204\),  
\(0>\)  \(0.9874>\)  \(0.9465>\)  \(0.9636>\)  \(0.9295>\)  \(0.9704>\)  \(0.9874>\)  
\(x_{5}\)  \(<1\),  \(<0.3444\),  \(<0.2611\),  \(<0.5667\),  \(<0.0833\),  \(<0.1944\),  
\(0>\)  \(0.9295>\)  \(0.9636>\)  \(0.6153>\)  \(0.9894>\)  \(0.9704>\)  
\(x_{6}\)  \(<1\),  \(<0.2911\),  \(<0.1244\),  \(<0.3444\),  \(<0.2704\),  
\(0>\)  \(0.9360>\)  \(0.9807>\)  \(0.9295>\)  \(0.9465>\)  
\(x_{7}\)  \(<1\),  \(<0.1800\),  \(<0.1500\),  \(<0.2611\),  
\(0>\)  \(0.9551>\)  \(0.9827>\)  \(0.9636>\)  
\(x_{8}\)  \(<1\),  \(<0.2333\),  \(<0.2333\),  
\(0>\)  \(0.9486>\)  \(0.9486>\)  
\(x_{9}\)  \(<1\),  \(<0.1944\),  
\(0>\)  \(0.9704>\)  
\(x_{10}\)  \(<1\),  
\(0>\) 
The computing result of \(R_{3}(U)\) from \(\mathrm{EC}_{3}\)
\(R_{3}(U)\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\)  \(x_{7}\)  \(x_{8}\)  \(x_{9}\)  \(x_{10}\) 

\(x_{1}\)  \(<1\),  \(<0.1111\),  \(<0\),  \(<0.7333\),  \(<0.2778\),  \(<0.4444\),  \(<0.8333\),  \(<0.5111\),  \(<0.2778\),  \(<0.6667\), 
\(0>\)  \(0.9809>\)  \(1>\)  \(0.3266>\)  \(0.9363>\)  \(0.6476>\)  \(0.2887>\)  \(0.6409>\)  \(0.9363>\)  \(0.3333>\)  
\(x_{2}\)  \(<1\),  \(<0.4444\),  \(<0.1778\),  \(<0.5370\),  \(<0.3704\),  \(<0.2778\),  \(<0.1037\),  \(<0.2704\),  \(<0.1111\),  
\(0>\)  \(0.6476>\)  \(0.9742>\)  \(0.6199>\)  \(0.6646>\)  \(0.9363>\)  \(0.9912>\)  \(0.9465>\)  \(0.9809>\)  
\(x_{3}\)  \(<1\),  \(<0.0667\),  \(<0.2778\),  \(<0.1111\),  \(<0.1667\),  \(<0.1778\),  \(<0.2778\),  \(<0\),  
\(0>\)  \(0.9933>\)  \(0.9363>\)  \(0.9809>\)  \(0.9553>\)  \(0.9742>\)  \(0.9363>\)  \(1>\)  
\(x_{4}\)  \(<1\),  \(<0.3444\),  \(<0.5111\),  \(<0.5667\),  \(<0.4578\),  \(<0.3444\),  \(<0.7333\),  
\(0>\)  \(0.9295>\)  \(0.6409>\)  \(0.6153>\)  \(0.6473>\)  \(0.9295>\)  \(0.3266>\)  
\(x_{5}\)  \(<1\),  \(<0.2037\),  \(<0.1944\),  \(<0.2704\),  \(<0.1204\),  \(<0.1944\),  
\(0>\)  \(0.9533>\)  \(0.9704>\)  \(0.9465>\)  \(0.9874>\)  \(0.9704>\)  
\(x_{6}\)  \(<1\),  \(<0.2778\),  \(<0.4370\),  \(<0.2037\),  \(<0.7778\),  
\(0>\)  \(0.9363>\)  \(0.6579>\)  \(0.9533>\)  \(0.3143>\)  
\(x_{7}\)  \(<1\),  \(<0.3444\),  \(<0.1944\),  \(<0.5000\),  
\(0>\)  \(0.9295>\)  \(0.9704>\)  \(0.6220>\)  
\(x_{8}\)  \(<1\),  \(<0.2704\),  \(<0.5111\),  
\(0>\)  \(0.9465>\)  \(0.6409>\)  
\(x_{9}\)  \(<1\),  \(<0.2778\),  
\(0>\)  \(0.9363>\)  
\(x_{10}\)  \(<1\),  
\(0>\) 
The computing result of \([x_{i}]_{\cap _{k = 1}^{3}R_{k}(U \times X)}\)
\([x_{i}]_{\cap _{k=1}^{3}R_{k}(U\times X)}\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\)  \(x_{7}\)  \(x_{8}\)  \(x_{9}\)  \(x_{10}\) 

\(x_{1}\)  \(<1\),  \(<0.1111\),  \(<0\),  \(<0.1750\),  \(<0.1667\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0>\)  \(0.9809>\)  \(1>\)  \(0.9615>\)  \(0.9553>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{2}\)  \(<0.1111\),  \(<1\),  \(<0.3444\),  \(<0.1778\),  \(<0.2333\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9809>\)  \(0>\)  \(0.9295>\)  \(0.9742>\)  \(0.9486>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{3}\)  \(<0\),  \(<0.3444\),  \(<1\),  \(<0.0667\),  \(<0.1750\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(1>\)  \(0.9295>\)  \(0>\)  \(0.9933>\)  \(0.9704>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{4}\)  \(<0.1750\),  \(<0.1778\),  \(<0.0667\),  \(<1\),  \(<0.1204\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9615>\)  \(0.9742>\)  \(0.9933>\)  \(0>\)  \(0.9874>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{5}\)  \(<0.1667\),  \(<0.2333\),  \(<0.1750\),  \(<0.1204\),  \(<1\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9553>\)  \(0.9486>\)  \(0.9704>\)  \(0.9874>\)  \(0>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{6}\)  \(<0.1778\),  \(<0.1244\),  \(<0.1111\),  \(<0.2704\),  \(<0.1750\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9742>\)  \(0.9807>\)  \(0.9809>\)  \(0.9465>\)  \(0.9615>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{7}\)  \(<0.2500\),  \(<0.1875\),  \(<0.1667\),  \(<0.2375\),  \(<0.1944\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9330>\)  \(0.9586>\)  \(0.9636>\)  \(0.9636>\)  \(0.9704>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{8}\)  \(<0.3750\),  \(<0.1037\),  \(<0.1778\),  \(<0.3000\),  \(<0.1250\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.7165>\)  \(0.9912>\)  \(0.9742>\)  \(0.9295>\)  \(0.9665>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  
\(x_{9}\)  \(<0.2778\),  \(<0.2333\),  \(<0.1944\),  \(<0.0725\),  \(<0.0833\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9363>\)  \(0.9486>\)  \(0.9704>\)  \(0.9919>\)  \(0.9894>\)  \(0>\)  \(1>\)  1  \(1>\)  \(1>\)  
\(x_{10}\)  \(<0.2778\),  \(<0.1111\),  \(<0\),  \(<0.1204\),  \(<0.0500\),  \(<0\),  \(<0\),  \(<0\),  \(<0\),  \(<0\), 
\(0.9363>\)  \(0.9809>\)  \(1>\)  \(0.9874>\)  \(0.9949>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\)  \(1>\) 
The results of uncertainty measures
No.  X  TypeI MGPFDTRSs  TypeII MGPFDTRSs  TypeIII MGPFDTRSs  TypeIV MGPFDTRSs  

\(\rho _{\mathrm{I}}\)  \(\sigma _{\mathrm{I}}\)  \(\omega _{\mathrm{I}}\)  \(\rho _{\mathrm{II}}\)  \(\sigma _{\mathrm{II}}\)  \(\omega _{\mathrm{II}}\)  \(\rho _{\mathrm{III}}\)  \(\sigma _{\mathrm{III}}\)  \(\omega _{\mathrm{III}}\)  \(\rho _{\mathrm{IV}}\)  \(\sigma _{\mathrm{IV}}\)  \(\omega _{\mathrm{IV}}\)  
1  \(x_{1,2,3,4,5}\)  0.7  1.4  0.7  0.2857  0.4  0.2  0.3  0.6  0.3  0.75  1.2  0.6 
2  \(x_{1,2,5,6,7}\)  0.7  1.4  0.7  0.5  0.6  0.3  0.3  0.6  0.3  0.75  1.2  0.6 
3  \(x_{1,3,5,7,8}\)  0.6  1.2  0.6  0.25  0.4  0.2  0.2  0.4  0.2  0.6667  1.2  0.6 
4  \(x_{1,3,4,6,7}\)  0.5  1  0.5  0.5  0.4  0.2  0.2  0.4  0.2  1  1  0.5 
5  \(x_{2,4,5,6,7}\)  0.7  1.4  0.7  0.5  0.6  0.3  0.4  0.8  0.4  1  1.2  0.6 
6  \(x_{2,4,5,9,10}\)  0.6  1.2  0.6  0.1429  0.2  0.1  0.1  0.2  0.1  0.875  1.4  0.6 
7  \(x_{2,7,8,9,10}\)  0.6  1.2  0.6  0  0  0  0.1  0.2  0.1  0.8571  1.2  0.6 
8  \(x_{3,4,6,8,9}\)  0.5  1  0.5  0.25  0.2  0.1  0.2  0.4  0.2  1  1  0.5 
9  \(x_{3,5,6,9,10}\)  0.6  1.2  0.6  0.2  0.2  0.1  0.1  0.2  0.1  0.8571  1.2  0.6 
10  \(x_{4,6,7,8,9}\)  0.5  1  0.5  0  0  0  0.1  0.2  0.1  1  1  0.5 
It is obvious that these four types of results are not entirely consistent. Then, the uncertainties are not entirely consistent for them and several kinds of uncertainty measure methods are necessary. Consequently, in different fields should select different model according the different requirements in practical application. Based on the calculated approximation sets, we can measure the uncertainty of the alternative MFs to estimate the investment. To evaluate the performance of the proposed uncertainty measure methods, we conduct a series of experiments to calculate these three uncertainty measures. Therefor, in our experiment, we randomly select five MFs from the set U, and uncertainty evolution’s for them as shown in Table 10.
According to Table 10 and Figs. 4, 5 and 6 we can get that the fifth set \(X = \{x_{2},x_{4},x_{5},x_{6},x_{7}\}\) with higher accuracy, approximation degree and approximation quality for each model. That is, \(x_{2}\), \(x_{4}\), \(x_{5}\), \(x_{6}\) and \(x_{7}\) are the best five MFs out of ten MFs for the investors to invest. There is no doubt that there are more recommended programs, but its best in the given ten options.
Conclusions
In this paper, we present four types of MGPFDTRSs of Pythagorean fuzzy subset (of a subset of the given universe) of the PFS (of the given universe) and study their uncertainty measure methods based on the Pythagorean fuzzy inclusion measure within the framework of multigranulation Pythagorean fuzzy approximation space. Using this four types of MGPFDTRSs and their uncertainty measure methods, we have presented a method for decisionmaking to IMSIS. In this decisionmaking method for IMSIS, we have analyzed three issues. (1) How to find the similarity degrees between two objects from IMSISs in the Pythagorean fuzzy settings. (2) How to obtain the Pythagorean fuzzy decisionmaking objects from IMSISs. (3) The following problem: if \(X_{1}, X_{2}, \ldots , X_{r} \subseteq U\) (U is the finite universe of discourse) then find the best \(X_{r}\), where the elements of \(X_{r}\) are randomly selected and not repeated any other elements in \(X_{r}\). The studies of this paper are focusing on the basis of the theoretical aspect and the general framework of decisionmaking process to the IMSIS of the proposed model and method. Therefore, it is recommended that the further improvement of the proposed method to apply more complexity decisionmaking problems in Garg [5, 6, 7, 9, 10], Mandal and Ranadive [27] and the reallife data be used to test the approach established in this paper.
Notes
Acknowledgements
The authors would like to thank the Guest Editor Harish Garg and reviewers for their thoughtful comments and valuable suggestions.
Compliance with ethical standards
Conflict of interest
Prasenjit Mandal and A. S. Ranadive declare that there is no conflict of interest.
Ethical approval
This article does not contain any study performed on humans or animals by the authors.
Informed consent
Informed consent was obtained from all individual participants included in the study.
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