Data Analysis Approach for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets
Abstract
:1. Introduction
2. Preliminaries
Relevant Definitions
3. Data Analysis Approaches for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets
3.1. Relevant Definitions
3.2. Data Analysis Approaches for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets
 (a)
 Input the incomplete intervalvalued intuitionistic fuzzy soft sets $(\tilde{\vartheta},E)$ and the parameter set E.
 (b)
 Find the missing degree of membership and nonmember ship of elements ${h}_{b}$ to $\tilde{\vartheta}\left({\epsilon}_{a}\right)$ as ${u}_{\tilde{\vartheta}(\epsilon )}^{*}(x)=[{u}_{\tilde{\vartheta}(\epsilon )}^{*}(x),{u}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)]$ (${v}_{\tilde{\vartheta}(\epsilon )}^{*}(x)=[{v}_{\tilde{\vartheta}(\epsilon )}^{*}(x),{v}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)]$).
 (c)
 Compute ${p}_{\tilde{\vartheta}({\epsilon}_{a})(u(\epsilon ))}^{*}$, ${p}_{\tilde{\vartheta}({\epsilon}_{a})(v(\epsilon ))}^{*}$, ${p}_{\tilde{\vartheta}({h}_{b})(u(\epsilon ))}^{*}$ and ${p}_{\tilde{\vartheta}({h}_{b})(v(\epsilon ))}^{*}$. If ${p}_{\tilde{\vartheta}({\epsilon}_{a})(u(\epsilon ))}^{*}\le 40\%\begin{array}{c}\left({p}_{\tilde{\vartheta}({\epsilon}_{a})(v(\epsilon ))}^{*}\le 40\%\right)\end{array}$, the remainder data which belong to the same column with the missing data are reliable; if ${p}_{\tilde{\vartheta}({h}_{b})(u(\epsilon ))}^{*}\le 40\%\begin{array}{c}({p}_{\tilde{\vartheta}({h}_{b})(v(\epsilon ))}^{*}\le 40\%\end{array})$, the remainder data which belong to the same row with the missing data are reliable; otherwise, this missing data should be ignored.
 (d)
 When the missing value is one of membership degree or nonmember ship degree, for $\forall x\in X$, $\mathrm{sup}{u}_{A}(x)+\mathrm{sup}{v}_{A}(x)\le 1$, we fill the missing data by the following equations:$$\{\begin{array}{l}\begin{array}{c}{u}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)=1{v}_{\tilde{\vartheta}(\epsilon )}^{+}(x),{u}_{\tilde{\vartheta}(\epsilon )}^{*}(x)={u}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)[{v}_{\tilde{\vartheta}(\epsilon )}^{+}(x){v}_{\tilde{\vartheta}(\epsilon )}^{}(x)]\end{array}\\ \begin{array}{c}{v}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)=1{u}_{\tilde{\vartheta}(\epsilon )}^{+}(x),{v}_{\tilde{\vartheta}(\epsilon )}^{*}(x)={v}_{\tilde{\vartheta}(\epsilon )}^{+*}(x)[{u}_{\tilde{\vartheta}(\epsilon )}^{+}(x){u}_{\tilde{\vartheta}(\epsilon )}^{}(x)]\end{array}\end{array}$$
 (e)
 When both membership degree and nonmember ship degree are missing, we calculate ${u}_{\tilde{\vartheta}({\epsilon}_{a})}^{*}({h}_{b})$ (${v}_{\tilde{\vartheta}({\epsilon}_{a})}^{*}({h}_{b})$ and ${u}_{\tilde{\vartheta}(a)}^{+*}({h}_{b})$ (${v}_{\tilde{\vartheta}(a)}^{+*}({h}_{b})$ as lower and upper (non) membership degrees of an element ${h}_{n}$ to $\tilde{\vartheta}({\epsilon}_{a})$, where
 (f)
 $$\{\begin{array}{l}{u}_{\tilde{\vartheta}({\epsilon}_{a})}^{*}({h}_{b})=({E}_{av{g}^{*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(u(\epsilon ))}+{\mathrm{H}}_{av{g}^{*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(u(\epsilon ))})/2\\ {u}_{\tilde{\vartheta}(a)}^{+*}({h}_{b})=({E}_{av{g}^{+*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(u(\epsilon ))}+{\mathrm{H}}_{av{g}^{+*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(u(\epsilon ))})/2\\ {v}_{\tilde{\vartheta}({\epsilon}_{a})}^{*}({h}_{b})=({E}_{av{g}^{*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(v(\epsilon ))}+{\mathrm{H}}_{av{g}^{*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(v(\epsilon ))})/2\\ {v}_{\tilde{\vartheta}(a)}^{+*}({h}_{b})=({E}_{av{g}^{+*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(v(\epsilon ))}+{\mathrm{H}}_{av{g}^{+*}{}_{\tilde{\vartheta}({\epsilon}_{a})}}{({h}_{b})}_{(v(\epsilon ))})/2\end{array}$$
 (g)
 Finally, we can get a complete intervalvalued intuitionistic fuzzy soft set.
3.3. One Example for the Proposed Approaches
4. Experimental Results
5. One RealLife Application
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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$\mathit{U}/\mathit{E}$  ${\mathit{e}}_{1}$  ${\mathit{e}}_{2}$  ${\mathit{e}}_{3}$  ${\mathit{e}}_{4}$ 

${h}_{1}$  [0.32,0.41],[0.39,0.52]  $[0.70,0.80],[0.10,0.20]$  $[0.75,0.85],[0.00,0.10]$  $[0.65,0.75],[0.10,0.20]$ 
${h}_{2}$  [0.50,0.82],[0.05,0.13]  $[0.60,0.80],[0.05,0.20]$  $[0.45,0.55],[0.20,0.40]$  $[0.60,0.70],[0.15,0.25]$ 
${h}_{3}$  [0.30,0.50],[0.20,0.40]  $[0.55,0.65],[0.20,0.30]$  [0.28,0.60],[0.22,0.35]  $[0.60,0.80],[0.05,0.15]$ 
${h}_{4}$  [0.73,0.85],[0.03,0.12]  $[0.75,0.85],[0.05,0.15]$  $[0.65,0.85],[0.05,0.15]$  [0.68,0.88],[0.01,0.12] 
${h}_{5}$  $[0.60,0.70],[0.10,0.20]$  [0.55,0.73],[0.11,0.23]  [0.62,0.75],[0.15,0.23]  $[0.40,0.50],[0.20,0.35]$ 
$$\mathit{U}/\mathit{E}$$

$${\mathit{\epsilon}}_{1}$$

$${\mathit{\epsilon}}_{2}$$

$${\mathit{\epsilon}}_{3}$$

$${\mathit{\epsilon}}_{4}$$

$${\mathit{\epsilon}}_{5}$$

$${\mathit{\epsilon}}_{6}$$


${h}_{1}$  [0.13,0.27] [0.60,0.67]  [0.77,0.88] [0.02,0.12]  [0.35,0.81] [0.05,0.15]  [0.14,0.40] [0.45,0.55]  [*,*] [0.60,0.67]  [0.70,0.80] [0.10,0.17] 
${h}_{2}$  [0.53,0.80] [0.00,0.20]  [0.60,0.80] [0.05,0.20]  [0.48,0.58] [0.20,0.30]  [0.38,0.69] [0.01,0.20]  [0.60,0.81] [0.09,0.19]  [0.38,0.69] [0.01,0.20] 
${h}_{3}$  [0.70,0.85] [0.05,0.15]  [0.70,0.89] [0.05,0.11]  [0.55,0.68] [0.20,0.30]  [0.76,0.87] [0.03,0.13]  [0.70,0.95] [0.01,0.05]  [0.67,0.80] [0.00,0.20] 
${h}_{4}$  [0.07,0.27] [0.60,0.67]  [0.69,0.95] [0.00,0.05]  [0.75,0.90] [0.01,0.10]  [0.41,0.88] [0.05,0.12]  [0.20,0.40] [0.00,0.40]  [0.75,0.85] [0.05,0.13] 
${h}_{5}$  [0.61,0.78] [0.11,0.19]  [0.65,0.75] [0.10,0.20]  [*,*] [*,*]  [0.39,0.71] [0.11,0.21]  [0.73,0.94] [0.01,0.06]  [*,*] [0.00,0.20] 
${h}_{6}$  [0.27,0.40] [0.00,0.53]  [0.65,0.85] [0.05,0.15]  [0.38,0.69] [0.01,0.20]  [0.70,0.80] [0.08,0.18]  [0.35,0.50] [0.20,0.40]  [0.23,0.26] [0.60,0.71] 
${h}_{7}$  [0.67,0.80] [0.00,0.20]  [0.74,1.00] [0.00,0.00]  [0.60,0.70] [0.15,0.25]  [0.70,0.89] [0.05,0.11]  [0.65,0.69] [0.21,0.31]  [0.69,0.80] [*,*] 
${h}_{8}$  [*,*] [0.60,0.73]  [0.72,0.82] [0.06,0.17]  [0.35,0.55] [0.30,0.40]  [*,*] [*,*]  [0.78,0.90] [0.01,0.10]  [0.70,0.95] [0.00,0.05] 
${h}_{9}$  [0.00,0.13] [0.60,0.80]  [0.71,0.81] [0.09,0.19]  [0.65,0.85] [0.05,0.15]  [0.23,0.71] [0.18,0.28]  [0.68,0.81] [0.08,0.18]  [0.20,0.40] [0.40,0.60] 
${h}_{10}$  [0.07,0.27] [0.33,0.60]  [0.58,0.68] [0.15,0.25]  [0.71,0.84] [0.05,0.15]  [0.32,0.55] [0.33,0.44]  [0.70,0.95] [0.00,0.05]  [0.23,0.26] [0.60,0.71] 
$\mathit{U}/\mathit{E}$  ${\mathit{\epsilon}}_{1}$  ${\mathit{\epsilon}}_{2}$  ${\mathit{\epsilon}}_{3}$  ${\mathit{\epsilon}}_{4}$  ${\mathit{\epsilon}}_{5}$  ${\mathit{\epsilon}}_{6}$ 

${h}_{1}$  [0.13,0.27] [0.60,0.67]  [0.77,0.88] [0.02,0.12]  [0.35,0.81] [0.05,0.15]  [0.14,0.40] [0.45,0.55]  [0.26,0.33] [0.60,0.67]  [0.70,0.80] [0.10,0.17] 
${h}_{2}$  [0.53,0.80] [0.00,0.20]  [0.60,0.80] [0.05,0.20]  [0.48,0.58] [0.20,0.30]  [0.38,0.69] [0.01,0.20]  [0.60,0.81] [0.09,0.19]  [0.38,0.69] [0.01,0.20] 
${h}_{3}$  [0.70,0.85] [0.05,0.15]  [0.70,0.89] [0.05,0.11]  [0.55,0.68] [0.20,0.30]  [0.76,0.87] [0.03,0.13]  [0.70,0.95] [0.01,0.05]  [0.67,0.80] [0.00,0.20] 
${h}_{4}$  [0.07,0.27] [0.60,0.67]  [0.69,0.95] [0.00,0.05]  [0.75,0.90] [0.01,0.10]  [0.41,0.88] [0.05,0.12]  [0.20,0.40] [0.00,0.40]  [0.75,0.85] [0.05,0.13] 
${h}_{5}$  [0.61,0.78] [0.11,0.19]  [0.65,0.75] [0.10,0.20]  [0.59,0.79] [0.11,0.20]  [0.39,0.71] [0.11,0.21]  [0.73,0.94] [0.01,0.06]  [0.6,0.80] [0.00,0.20] 
${h}_{6}$  [0.27,0.40] [0.00,0.53]  [0.65,0.85] [0.05,0.15]  [0.38,0.69] [0.01,0.20]  [0.70,0.80] [0.08,0.18]  [0.35,0.50] [0.20,0.40]  [0.23,0.26] [0.60,0.71] 
${h}_{7}$  [0.67,0.80] [0.00,0.20]  [0.74,1.00] [0.00,0.00]  [0.60,0.70] [0.15,0.25]  [0.70,0.89] [0.05,0.11]  [0.65,0.69] [0.21,0.31]  [0.69,0.80] [0.09,0.20] 
${h}_{8}$  [0.14,0.27] [0.60,0.73]  [0.72,0.82] [0.06,0.17]  [0.35,0.55] [0.30,0.40]  [0.51,0.72] [0.17,0.27]  [0.78,0.90] [0.01,0.10]  [0.70,0.95] [0.00,0.05] 
${h}_{9}$  [0.00,0.13] [0.60,0.80]  [0.71,0.81] [0.09,0.19]  [0.65,0.85] [0.05,0.15]  [0.23,0.71] [0.18,0.28]  [0.68,0.81] [0.08,0.18]  [0.20,0.40] [0.40,0.60] 
${h}_{10}$  [0.07,0.27] [0.33,0.60]  [0.58,0.68] [0.15,0.25]  [0.71,0.84] [0.05,0.15]  [0.32,0.55] [0.33,0.44]  [0.70,0.95] [0.00,0.05]  [0.23,0.26] [0.60,0.71] 
$\mathit{U}/\mathit{E}$  ${\mathit{\epsilon}}_{1}$  ${\mathit{\epsilon}}_{2}$  ${\mathit{\epsilon}}_{3}$  ${\mathit{\epsilon}}_{4}$  ${\mathit{\epsilon}}_{5}$ 

${h}_{1}$  [0.5,0.7] [0.1,0.2]  [0.8,0.9] [0.0,0.1]  [0.4,0.6] [0.2,0.3]  [0.5,0.6] [0.2,0.3]  [0.7,0.9] [0.0,0.1] 
${h}_{2}$  [0.6,0.7] [0.1,0.2]  [0.4,0.5] [0.2,0.4]  [0.6,0.8] [0.1,0.2]  [0.6,0.8] [0.0,0.2]  [0.6,0.8] [0.1,0.2] 
${h}_{3}$  [0.2,0.5] [0.1,0.3]  [0.6,0.7] [0.1,0.2]  [0.4,0.6] [0.3,0.4]  [0.7,0.9] [0.0,0.1]  [0.7,0.8] [0.1,0.2] 
${h}_{4}$  [0.5,0.6] [0.1,0.4]  [0.6,0.8] [0.0,0.1]  [0.7,0.8] [0.1,0.2]  [0.7,0.8] [0.1,0.2]  [0.6,0.9] [0.0,0.1] 
${h}_{5}$  [0.5,0.7] [0.1,0.2]  [0.7,0.9] [0.0,0.1]  [0.5,0.6] [0.1,0.2]  [0.6,0.8] [0.1,0.2]  [0.5,0.8] [0.1,0.2] 
${h}_{6}$  [0.7,0.8] [0.1,0.2]  [0.7,0.9] [0.0,0.1]  [0.6,0.8] [0.1,0.2]  [0.7,0.8] [0.1,0.2]  [0.6,0.8] [0.0,0.1] 
${h}_{7}$  [0.4,0.8] [0.1,0.2]  [0.6,0.7] [0.1,0.2]  [0.7,0.8] [0.1,0.2]  [0.6,0.8] [0.1,0.2]  [0.6,0.7] [0.1,0.2] 
${h}_{8}$  [0.3,0.7] [0.1,0.3]  [0.6,0.8] [0.1,0.2]  [0.5,0.7] [0.2,0.3]  [0.8,0.9] [0.0,0.1]  [0.8,0.9] [0.0,0.1] 
${h}_{9}$  [0.5,0.6] [0.2,0.3]  [0.8,0.9] [0.0,0.1]  [0.7,0.9] [0.0,0.1]  [0.7,0.8] [0.1,0.2]  [0.6,0.8] [0.1,0.2] 
${h}_{10}$  [0.6,0.7] [0.2,0.3]  [0.7,0.8] [0.1,0.2]  [0.7,0.8] [0.1,0.2]  [0.7,0.9] [0.0,0.1]  [0.6,0.9] [0.0,0.1] 
$\mathit{U}/\mathit{E}$  ${\mathit{\epsilon}}_{1}$  ${\mathit{\epsilon}}_{2}$  ${\mathit{\epsilon}}_{3}$  ${\mathit{\epsilon}}_{4}$  ${\mathit{\epsilon}}_{5}$ 

${h}_{1}$  [0.25,0.34] [0.50,0.66]  [0.17,0.25] [0.50,0.70]  [0.20,0.41] [0.30,0.50]  [*,*] [*,*]  [0.11, 0.29] [0.60,0.70] 
${h}_{2}$  [0.55,0.68] [0.10,0.30]  [0.61,0.80] [0.10,0.20]  [0.42,0.61] [0.10,0.33]  [0.75,0.80] [0.01,0.20]  [0.20,0.40] [0.40,0.60] 
${h}_{3}$  [0.15,0.45] [0.22,0.55]  [0.30,0.40] [0.20,0.50]  [0.25,0.60] [0.22,0.40]  [0.36,0.50] [0.21,0.50]  [0.70,0.85] [0.00,0.10] 
${h}_{4}$  [0.07,0.27] [0.60,0.67]  [0.20,0.35] [0.40,0.60]  [0.60,0.71] [0.01,0.20]  [0.40,0.80] [0.02,0.12]  [0.30,0.40] [0.30,0.50] 
${h}_{5}$  [0.31,0.48] [0.22,0.50]  [0.70,0.90] [0.00,0.10]  [0.32, 0.42] [0.20, 0.50]  [0.61,0.71] [0.10,0.25]  [0.30,0.60] [0.11,0.32] 
${h}_{6}$  [0.40,0.60] [0.11,0.33]  [0.41,0.72] [0.01,0.22]  [0.10,0.32] [0.42,0.60]  [0.72,0.80] [0.10,0.20]  [0.40,0.55] [0.20,0.45] 
${h}_{7}$  [0.60,0.70] [*,*]  [0.50,0.071 [0.06,0.22]  [0.50,1.00] [0.00,0.00]  [0.70,0.90] [0.00,0.10]  [0.50,0.70] [0.11,0.30] 
$\mathit{U}/\mathit{E}$  ${\mathit{\epsilon}}_{1}$  ${\mathit{\epsilon}}_{2}$  ${\mathit{\epsilon}}_{3}$  ${\mathit{\epsilon}}_{4}$  ${\mathit{\epsilon}}_{5}$ 

${h}_{1}$  [0.25,0.34] [0.50,0.66]  [0.17,0.25] [0.50,0.70]  [0.20,0.41] [0.30,0.50]  [0.39,0.54] [0.28,0.44]  [0.11, 0.29] [0.60,0.70] 
${h}_{2}$  [0.55,0.68] [0.10,0.30]  [0.61,0.80] [0.10,0.20]  [0.42,0.61] [0.10,0.33]  [0.75,0.80] [0.01,0.20]  [0.20,0.40] [0.40,0.60] 
${h}_{3}$  [0.15,0.45] [0.22,0.55]  [0.30,0.40] [0.20,0.50]  [0.25,0.60] [0.22,0.40]  [0.36,0.50] [0.21,0.50]  [0.70,0.85] [0.00,0.10] 
${h}_{4}$  [0.07,0.27] [0.60,0.67]  [0.20,0.35] [0.40,0.60]  [0.60,0.71] [0.01,0.20]  [0.40,0.80] [0.02,0.12]  [0.30,0.40] [0.30,0.50] 
${h}_{5}$  [0.31,0.48] [0.22,0.50]  [0.70,0.90] [0.00,0.10]  [0.32, 0.42] [0.20, 0.50]  [0.61,0.71] [0.10,0.25]  [0.30,0.60] [0.11,0.32] 
${h}_{6}$  [0.40,0.60] [0.11,0.33]  [0.41,0.72] [0.01,0.22]  [0.10,0.32] [0.42,0.60]  [0.72,0.80] [0.10,0.20]  [0.40,0.55] [0.20,0.45] 
${h}_{7}$  [0.60,0.70] [0.20,0.30]  [0.50,0.071 [0.06,0.22]  [0.50,1.00] [0.00,0.00]  [0.70,0.90] [0.00,0.10]  [0.50,0.70] [0.11,0.30] 
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Qin, H.; Li, H.; Ma, X.; Gong, Z.; Cheng, Y.; Fei, Q. Data Analysis Approach for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets. Symmetry 2020, 12, 1061. https://doi.org/10.3390/sym12071061
Qin H, Li H, Ma X, Gong Z, Cheng Y, Fei Q. Data Analysis Approach for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets. Symmetry. 2020; 12(7):1061. https://doi.org/10.3390/sym12071061
Chicago/Turabian StyleQin, Hongwu, Huifang Li, Xiuqin Ma, Zhangyun Gong, Yuntao Cheng, and Qinghua Fei. 2020. "Data Analysis Approach for Incomplete IntervalValued Intuitionistic Fuzzy Soft Sets" Symmetry 12, no. 7: 1061. https://doi.org/10.3390/sym12071061