# Incomplete Fuzzy Soft Sets and Their Application to Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

## 3. Incomplete Fuzzy Soft Set Based Decision-Making

Algorithm 1 for incomplete fuzzy soft set based decision making: |

Step 1: Consider the incomplete fuzzy soft set $(f,E)$, where $u\in U$, $e\in E$.Step 2: Give a threshold fuzzy set $E:S\to [0,1]$ and compute the level soft set $({f}_{\lambda},E)$ with respect to the threshold fuzzy set $\lambda $.Step 3: Present the level soft set $({f}_{\lambda},E)$ in tabular form, where missing data are still denoted by the sign “*”.Step 4: Compute the choice value ${c}_{i}$ of ${u}_{i}$ according to $({f}_{\lambda},E)$, where ${c}_{i}={\sum}_{j}{f}_{\lambda}\left({u}_{i}\right)\left({e}_{j}\right)$.Remove each row with the smallest choice value ${c}_{i}$ (Whether * takes 0 or 1, the choice value ${c}_{i}$ is the smallest). The number of remaining objects is denoted by s, and the number of attributes from E is denoted by t. The reduced incomplete soft set is denoted by $({f}_{\lambda}^{\prime},E)$. Step 5: In the modified $s\times t$ matrix, list the cells with value * as $(({u}_{{i}_{1}},{e}_{{j}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}}))$.Step 6: By every vector $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{w})\in {\{0,1\}}^{w}$, we construct a $s\times t$ matrix ${P}_{\alpha}={\left({p}_{{u}_{i},{e}_{j}}\right)}_{s\times t}$ where
- (1)
- ${p}_{{u}_{i},{e}_{j}}={f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)$ if $({u}_{i},{e}_{j})$ is not listed in $(({u}_{{i}_{1}},{e}_{{j}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}})),$ where ${f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)={f}_{\lambda}\left({e}_{j}\right)\left({u}_{i}\right)$.
- (2)
- ${p}_{{u}_{i},{e}_{j}}={\alpha}_{v}$ if $({u}_{i},{e}_{j})=({u}_{{i}_{v}},{e}_{{j}_{v}})$, $v\in \{1,2,\cdots ,w\}$.
Step 7: For every ${u}_{i}$, let ${n}_{{u}_{i}}$ regarded as the number of vectors $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{w})\in {\{0,1\}}^{w}$ for which object ${u}_{i}$ maximizes the choice value at ${P}_{\alpha}$. Let ${o}_{{u}_{i}}={n}_{{u}_{i}}/{2}^{w}$. Define ${o}_{{u}_{i}}=0$ for dominated alternatives.Step 8: The optimal decision is to choose ${u}_{l}$ satisfying ${o}_{{u}_{l}}=ma{x}_{i=1\cdots s}{o}_{{u}_{i}}$.Step 9: If l has multiple values, then any one may be selected. |

**Example**

**1.**

## 4. Weighted Incomplete Fuzzy Soft Sets Based Decision Making

**Definition**

**6**

**Definition**

**7.**

Algorithm 2 for weighted incomplete fuzzy soft set based decision-making: |

Step 1: Consider the weighted incomplete fuzzy soft set $(f,E,W)$, where $u\in U$, $e\in E$.Step 2: Give a threshold fuzzy set $\lambda :E\to [0,1]$ for decision-making and compute the level soft set $({f}_{\lambda},E,W)$ with respect to the threshold fuzzy set $\lambda $.Step 3: Present the level soft set $({f}_{\lambda},E,W)$ in tabular form, where missing data are still denoted by the sign “*”.Step 4: Compute the weighted choice value ${c}_{i}$ of ${u}_{i}$ according to $({f}_{\lambda},E,W)$, where ${c}_{i}={\sum}_{j}({f}_{\lambda}\left({u}_{i}\right)\left({e}_{j}\right)\times {W}_{j})$.Remove each row with the smallest weighted choice value ${c}_{i}$ (whether * takes 0 or 1, the choice value ${c}_{i}$ is the smallest).The number of remaining objects is denoted by s, and the number of attributes from E is denoted by t. The reduced weighted incomplete soft set is denoted by $({f}_{\lambda}^{\prime},E,W)$ Step 5: In the modified $s\times t$ matrix, list the cells with value * as $(({u}_{{i}_{1}},{e}_{{j}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}}))$.Step 6: By every vector $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{w})\in {\{0,1\}}^{w}$, we construct a $s\times t$ weighted matrix ${P}_{\alpha}={\left({p}_{{u}_{i},{e}_{j}}\right)}_{s\times t}$ where- (1)
- ${p}_{{u}_{i},{e}_{j}}={f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)$ if $({u}_{i},{e}_{j})$ is not listed in $(({u}_{{i}_{1}},{e}_{{j}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}}))$, where ${f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)={f}_{\lambda}\left({e}_{j}\right)\left({u}_{i}\right)$.
- (2)
- ${p}_{{u}_{i},{e}_{j}}={\alpha}_{v}$ if $({u}_{i},{e}_{j})=({u}_{{i}_{v}},{e}_{{j}_{v}})$, $v\in \{1,2,\cdots ,w\}$.
Step 7: For every ${u}_{i}$, let ${n}_{{u}_{i}}$ be regarded as the number of vectors $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{w})\in {\{0,1\}}^{w}$ for which object ${u}_{i}$ maximizes the weighted choice value at ${P}_{\alpha}$. Let ${o}_{{u}_{i}}={n}_{{u}_{i}}/{2}^{w}$. Define ${o}_{{u}_{i}}=0$ for dominated alternatives.Step 8: The optimal decision is to choose ${u}_{l}$ if ${o}_{{u}_{l}}=ma{x}_{i=1\cdots s}{o}_{{u}_{i}}$.Step 9: If l has multiple values, then any one may be selected. |

## 5. Incomplete Weighted Fuzzy Soft Sets Based Decision Making

**Definition**

**8.**

Algorithm 3 for incomplete weighted fuzzy soft set based decision-making: |

Step 1: Considering incomplete weighted fuzzy soft set $(f,E,\tilde{W})$, where $u\in U$, $e\in E$.Step 2: Give a threshold fuzzy set $\lambda :E\to [0,1]$ for decision-making and compute the level soft set $({f}_{\lambda},E,\tilde{W})$ with respect to the threshold fuzzy set $\lambda $.Step 3: Present the level soft set $({f}_{\lambda},E)$ in tabular form, where missing data $\left({f}_{\lambda}\left({u}_{i}\right)\left({e}_{j}\right)\right)$ are still denoted by the sign “*” and missing weighted values are still marked as the sign “★”.Step 4: Compute the incomplete weighted choice value ${c}_{i}$ of ${u}_{i}$ according to $({f}_{\lambda},E,\tilde{W})$, where ${c}_{i}={\sum}_{j}({f}_{\lambda}\left({u}_{i}\right)\left({e}_{j}\right)\times {W}_{j})$.Remove each row with the smallest incomplete weighted choice value ${c}_{i}$. (Whether * and ★ takes 0 or 1, the choice value ${c}_{i}$ is the smallest). The number of remaining objects is denoted by s, and the number of attributes from E is denoted by t. The reduced incomplete weighted soft set is denoted by $({f}_{\lambda}^{\prime},E,\tilde{W}).$ Step 5: In the modified $s\times t$ matrix, list the cells with value * as $(({u}_{{i}_{1}},{e}_{{j}_{1}},\tilde{{W}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}},\tilde{{W}_{w}}))$.Step 6: By every vector $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{w})\in {\{0,1\}}^{w}$, we construct a $s\times t$ incomplete weighted matrix ${P}_{\alpha}={\left({p}_{{u}_{i},{e}_{j}}\right)}_{s\times t}$ where- (1)
- ${p}_{{u}_{i},{e}_{j}}={f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)$ if $({u}_{i},{e}_{j})$ is not listed in $(({u}_{{i}_{1}},{e}_{{j}_{1}}),\cdots ,({u}_{{i}_{w}},{e}_{{j}_{w}})),$ where ${f}_{\lambda}^{-1}\left({u}_{i}\right)\left({e}_{j}\right)={f}_{\lambda}\left({e}_{j}\right)\left({u}_{i}\right)$.
- (2)
- ${p}_{{u}_{i},{e}_{j}}={\alpha}_{v}$ if $({u}_{i},{e}_{j})=({u}_{{i}_{v}},{e}_{{j}_{v}})$, $v\in \{1,2,\cdots ,w\}$.
Step 7: By computing the incomplete weighted choice value ${c}_{{\alpha}_{i}}$ of each ${P}_{\alpha}={\left({p}_{{u}_{i},{e}_{j}}\right)}_{s\times t}$, we construct a new $s\times \alpha $ matrix $\tilde{P}=({\tilde{p}}_{{u}_{i},{c}_{{\alpha}_{i}}})$ where each column of the matrix $\tilde{P}$ represents the incomplete weighted choice value of each ${P}_{\alpha}$ for every object. Missing weighted values are still marked as the sign “★”. (The weighted choice value containing unknown data is regarded as an incomplete weighted choice value.)Step 8: Get the maximum choice value ${C}_{\alpha}$ without “★” and the maximum choice value ${C}_{\alpha}^{\prime}$ with “★” in each column of the matrix. Compute the critical values ${\tilde{C}}^{\alpha}=|{C}_{\alpha}-{C}_{\alpha}^{\prime}|$ in each column of the matrix where ★ is regarded as 0. Let ${\tilde{C}}^{\alpha}$ remove duplicate values and get ${\tilde{C}}^{q}$ where q is the number of ${\tilde{C}}^{q}$ .Step 9: Sorting and classifying each critical value ${\tilde{C}}^{q}$ such that we get $2q+1$ matrices ${R}^{k}$ in different intervals where $k=\{1,\cdots ,2q+1\}$.Step 10: For every ${u}_{i}$, let ${n}_{{u}_{i}}$ be regarded as the number whose object ${u}_{i}$ maximizes the weighted choice value at ${R}^{k}$. Select the object corresponding to the maximum number ${n}_{{u}_{i}}$ in each interval. Summarize and merge the interval values belonging to the same object.Step 11: In the object corresponding to the modified interval value, the object with the most common attributes is chosen as the optimal decision ${u}_{l}$.Step 12: If l has multiple values, then any one may be selected. |

## 6. Algorithm of Incomplete Fuzzy Soft Sets Associated with a Modal-Style Operator

Algorithm 4 for incomplete fuzzy soft set based decision-making by using modal-style operators: |

Step 1: Considering the incomplete fuzzy soft set $(f,E)$, where $u\in U$, $e\in E$.Step 2: All unknown data are replaced with 0 and input the optimum decision ${\theta}_{0}\in f\left(E\right)$ given by ${\theta}_{0}\left({e}_{i}\right)=max\left\{f\left({u}_{i}\right)\left({e}_{j}\right)|{u}_{i}\in U,{e}_{j}\in E\right\}$.Step 3: Calculate ${\theta}_{0}^{\downarrow}$ and ${\theta}_{0}^{\diamond}$.Step 4: All unknown data are replaced with 1 and input the optimum decision ${\theta}_{1}\in f\left(E\right)$ given by ${\theta}_{1}\left({e}_{i}\right)=max\left\{f\left({u}_{i}\right)\left({e}_{j}\right)|{u}_{i}\in U,{e}_{j}\in E\right\}$.Step 5: Calculate ${\theta}_{1}^{\downarrow}$ and ${\theta}_{1}^{\diamond}$.Step 6: Calculate the choice value $\sigma \left(u\right)$ for each $u\in U$, where $\sigma \left(u\right)=\frac{{\theta}_{0}^{\downarrow}\left(u\right)+{\theta}_{1}^{\downarrow}\left(u\right)+{\theta}_{0}^{\diamond}\left(u\right)+{\theta}_{1}^{\diamond}\left(u\right)}{2}$.Step 7: The optimal decision is to choose $l\in U$ satisfying $\sigma \left(l\right)=ma{x}_{u\in U}\sigma \left(u\right)$.Step 8: If l has multiple values, then any one may be selected. |

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{s}}_{6}$ | |
---|---|---|---|---|---|---|

${v}_{1}$ | 1 | 1 | 0 | 0 | 0 | 0 |

${v}_{2}$ | * | 0 | 1 | 0 | 1 | 0 |

${v}_{3}$ | 1 | * | 1 | 0 | 0 | 1 |

${v}_{4}$ | 0 | 0 | 0 | 1 | 0 | 1 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{s}}_{6}$ | |
---|---|---|---|---|---|---|

${v}_{1}$ | 0.9 | 0.4 | 0.5 | 0.4 | 0.8 | 0.8 |

${v}_{2}$ | 0.8 | * | 0.5 | 0.7 | 0.6 | 0.3 |

${v}_{3}$ | 0.4 | * | 0.9 | 0.9 | 0.5 | 0.9 |

${v}_{4}$ | 0.9 | 0.8 | 0.9 | 0.4 | 0.7 | 0.5 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | |
---|---|---|---|---|---|

${v}_{1}$ | 0.1 | 0.5 | 0.3 | 0.4 | 0.3 |

${v}_{2}$ | * | 0.5 | 0.2 | 0.3 | 0.6 |

${v}_{3}$ | 0.1 | * | 0.4 | 0.5 | 0.1 |

${v}_{4}$ | 0.7 | 0.2 | 0.2 | 0.2 | 0.3 |

${v}_{5}$ | 0.2 | 0.6 | 0.3 | * | 0.3 |

${v}_{6}$ | 0.9 | 0.2 | 0.1 | 0.1 | 0.8 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | |
---|---|---|---|---|---|

${v}_{1}$ | 0 | 1 | 0 | 0 | 0 |

${v}_{2}$ | * | 1 | 0 | 0 | 1 |

${v}_{3}$ | 0 | * | 0 | 1 | 0 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 |

${v}_{5}$ | 0 | 1 | 0 | * | 0 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | |
---|---|---|---|---|---|

${v}_{2}$ | * | 1 | 0 | 0 | 1 |

${v}_{3}$ | 0 | * | 0 | 1 | 0 |

${v}_{5}$ | 0 | 1 | 0 | * | 0 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 1 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 1 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 3 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 1 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 1 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 1 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 1 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 3 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 1 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 3 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 1 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 3 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 2 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 2 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | |
---|---|---|---|---|---|

${v}_{1}$ | 0.1 | 0.5 | 0.3 | 0.4 | 0.3 |

${v}_{2}$ | * | 0.5 | 0.2 | 0.3 | 0.6 |

${v}_{3}$ | 0.1 | * | 0.4 | 0.5 | 0.1 |

${v}_{4}$ | 0.7 | 0.2 | 0.2 | 0.2 | 0.3 |

${v}_{5}$ | 0.2 | 0.6 | 0.3 | * | 0.3 |

${v}_{6}$ | 0.9 | 0.2 | 0.1 | 0.1 | 0.8 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | |
---|---|---|---|---|---|

${v}_{1}$ | 0 | 1 | 0 | 0 | 0 |

${v}_{2}$ | * | 1 | 0 | 0 | 1 |

${v}_{3}$ | 0 | * | 0 | 1 | 0 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 |

${v}_{5}$ | 0 | 1 | 0 | * | 0 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | |
---|---|---|---|---|---|

${v}_{2}$ | * | 1 | 0 | 0 | 1 |

${v}_{3}$ | 0 | * | 0 | 1 | 0 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 |

${v}_{5}$ | 0 | 1 | 0 | * | 0 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 1.1 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 0.7 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 0.6 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 0.7 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 0.6 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 1.1 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 0.6 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 1.1 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 0.7 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 0 | 0 | 0.6 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 0 | 0 | 1 | 0 | 0.7 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 0 | 1 | 0 | 0 | 1 | 1.1 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=0.6$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | ${\mathit{c}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

${v}_{2}$ | 1 | 1 | 0 | 0 | 1 | 2 |

${v}_{3}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 | 0.9 |

${v}_{5}$ | 0 | 1 | 0 | 1 | 0 | 1.3 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 | 1.4 |

${\mathit{s}}_{1},{\tilde{\mathit{W}}}_{1}=0.9$ | ${\mathit{s}}_{2},{\tilde{\mathit{W}}}_{2}=\u2605$ | ${\mathit{s}}_{3},{\tilde{\mathit{W}}}_{3}=0.6$ | ${\mathit{s}}_{4},{\tilde{\mathit{W}}}_{4}=0.7$ | ${\mathit{s}}_{5},{\tilde{\mathit{W}}}_{5}=0.5$ | |
---|---|---|---|---|---|

${v}_{1}$ | 0.1 | 0.5 | 0.3 | 0.4 | 0.3 |

${v}_{2}$ | * | 0.5 | 0.2 | 0.3 | 0.6 |

${v}_{3}$ | 0.1 | * | 0.4 | 0.5 | 0.1 |

${v}_{4}$ | 0.7 | 0.2 | 0.2 | 0.2 | 0.3 |

${v}_{5}$ | 0.2 | 0.6 | 0.3 | * | 0.3 |

${v}_{6}$ | 0.9 | 0.2 | 0.1 | 0.1 | 0.8 |

${\mathit{s}}_{1},{\mathit{W}}_{1}=0.9$ | ${\mathit{s}}_{2},{\mathit{W}}_{2}=\u2605$ | ${\mathit{s}}_{3},{\mathit{W}}_{3}=0.6$ | ${\mathit{s}}_{4},{\mathit{W}}_{4}=0.7$ | ${\mathit{s}}_{5},{\mathit{W}}_{5}=0.5$ | |
---|---|---|---|---|---|

${v}_{2}$ | * | 1 | 0 | 0 | 1 |

${v}_{3}$ | 0 | * | 0 | 1 | 0 |

${v}_{4}$ | 1 | 0 | 0 | 0 | 0 |

${v}_{5}$ | 0 | 1 | 0 | * | 0 |

${v}_{6}$ | 1 | 0 | 0 | 0 | 1 |

${\mathit{c}}_{{\mathit{\alpha}}_{1}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{2}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{3}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{4}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{5}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{6}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{7}}$ | ${\mathit{c}}_{{\mathit{\alpha}}_{8}}$ | |
---|---|---|---|---|---|---|---|---|

${v}_{2}$ | $0.5+\u2605$ | $1.4+\u2605$ | $0.5+\u2605$ | $0.5+\u2605$ | $1.4+\u2605$ | $1.4+\u2605$ | $0.5+\u2605$ | $1.4+\u2605$ |

${v}_{3}$ | 0.7 | 0.7 | $0.7+\u2605$ | 0.7 | $0.7+\u2605$ | 0.7 | $0.7+\u2605$ | $0.7+\u2605$ |

${v}_{4}$ | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 |

${v}_{5}$ | ★ | ★ | ★ | $0.7+\u2605$ | ★ | $0.7+\u2605$ | $0.7+\u2605$ | $0.7+\u2605$ |

${v}_{6}$ | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 |

${\mathit{n}}_{{\mathit{v}}_{2}}$ | ${\mathit{n}}_{{\mathit{v}}_{3}}$ | ${\mathit{n}}_{{\mathit{v}}_{4}}$ | ${\mathit{n}}_{{\mathit{v}}_{5}}$ | ${\mathit{n}}_{{\mathit{v}}_{6}}$ | |
---|---|---|---|---|---|

$0\le \u2605<0.7$ | 4 | 0 | 0 | 0 | 4 |

$0.7<\u2605<0.9$ | 4 | 2 | 0 | 2 | 0 |

$0.9<\u2605\le 1$ | 5 | 2 | 0 | 2 | 0 |

$\u2605=0.7$ | 4 | 2 | 0 | 2 | 4 |

$\u2605=0.9$ | 5 | 2 | 0 | 2 | 1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, L.; Qin, K.
Incomplete Fuzzy Soft Sets and Their Application to Decision-Making. *Symmetry* **2019**, *11*, 535.
https://doi.org/10.3390/sym11040535

**AMA Style**

Wang L, Qin K.
Incomplete Fuzzy Soft Sets and Their Application to Decision-Making. *Symmetry*. 2019; 11(4):535.
https://doi.org/10.3390/sym11040535

**Chicago/Turabian Style**

Wang, Lu, and Keyun Qin.
2019. "Incomplete Fuzzy Soft Sets and Their Application to Decision-Making" *Symmetry* 11, no. 4: 535.
https://doi.org/10.3390/sym11040535