# A Decision-Making Algorithm Based on the Average Table and Antitheses Table for Interval-Valued Fuzzy Soft Set

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary and Related Work

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- U is the set including six car candidates, and then U = {h
_{1}, h_{2}, h_{3}, h_{4}, h_{5}, h_{6}}; - A is the set of parameters, and A = {ε
_{1}, ε_{2}, ε_{3}, ε_{4}, ε_{6}} = {power, cheap, security, ride comfort, braking performance}.

_{1}, while the degree of the power of the car h

_{1}is at least 0.3, and at most 0.5.

- Give the interval-valued fuzzy soft set (F, A).
- $\forall $h
_{i}$\in $ U, figure out the choice value c_{i}for each object h_{i}such that ${c}_{i}=[{c}_{i}^{-},{c}_{i}^{+}]=[{\displaystyle \sum _{p\in P}{\mu}_{\tilde{H}(p)}^{-}({h}_{i})},{\displaystyle \sum _{p\in P}{\mu}_{\tilde{H}(p)}^{+}({h}_{i})}]$. - $\forall $h
_{i}$\in $ U, achieve the score r_{i}of h_{i}such that ${r}_{i}={\displaystyle \sum _{{h}_{j}\in U}(({c}_{i}^{-}-{c}_{j}^{-})+({c}_{i}^{+}-{c}_{j}^{+}))}$. - Choose any one of the objects h
_{k}$\in $ U such that ${r}_{k}={\mathrm{max}}_{{h}_{i}\in U}\left\{{r}_{i}\right\}$ as the best candidate.

_{i}and the score r

_{i}for all h

_{i}$\in $ U. In the end, the relevant outcome is shown in Table 2, from which we can conclude that h

_{5}is the best option, in line with Yang et al.’s algorithm, because it has the maximum score r

_{5}= 6.1.

- Input the (resultant) interval-valued fuzzy soft set $(\tilde{F},A)$ and an opinion weighting vector $W=(\alpha ,\beta )$;
- Work out Weighted Reduct Fuzzy Soft Set(WRFS) ${\mathsf{\Phi}}_{W}=({\tilde{F}}_{W},A)$ of $(\tilde{F},A)$ with respect to $W$;
- Choose an aggregation operation G;
- Figure out and display the level soft set $L({\mathsf{\Phi}}_{W};G)$ in tables;
- Work out the choice value c
_{i}of o_{i}, $\forall $i; - Select h
_{k}if ${c}_{k}={\mathrm{max}}_{i}{c}_{i}$ as the optimal choice.

_{5}is the largest, and the maximum value is 3. Based on this result, the car buyer ought to choose h

_{5}as the best option, in line with Feng et al.’s algorithm.

## 3. The Proposed Decision-Making Algorithm

#### 3.1. The Related Definitions

**Definition**

**4.**

**Definition**

**5.**

_{ij}in the table is defined as the sum of the non-negative values of the below-definedlimited list:

_{j}(j = 1, 2, 3, …, m) is the maximum mean membership value for each parameter in each column in the average table.

**Definition**

**6.**

_{i}of an object h

_{i}is as follows:

_{j}of an object h

_{j}can be calculated as follows:

**Definition**

**7.**

_{i}is S

_{i}, which may be computed as follows:

#### 3.2. The Proposed Algorithm

- Input the interval-valued fuzzy soft set (F, A).
- Obtain the average table, in which entry is denoted as a
_{ij}, by calculating the mean degree of membership given by the above definition. - Find the maximum mean membership value, called Q
_{j}(j = 1, 2, 3, …, m), for each parameter in each column in the average table. - Construct the antitheses table. Each element b
_{ij}in the table is defined as the sum of the non-negative values of the below-mentioned limited list:$\frac{{\mathrm{a}}_{i1}-{\mathrm{a}}_{j1}}{{Q}_{1}}$,$\frac{{\mathrm{a}}_{i2}-{\mathrm{a}}_{j2}}{{Q}_{2}}$,$\frac{{\mathrm{a}}_{i3}-{\mathrm{a}}_{j3}}{{Q}_{3}}$,……,$\frac{{\mathrm{a}}_{im}-{\mathrm{a}}_{jm}}{{Q}_{m}}$ - Compute the row-sum M
_{i}and column-sum N_{i}in the antitheses table and the score S_{i}for each object h_{i}, i = 1, 2, 3, …, n. - The final decision is any object h
_{k}which has the highest score value, i.e., any h_{k}such that S_{k}= maxi S_{i}.

#### 3.3. Example

_{1}, h

_{2}, h

_{3}, h

_{4}, h

_{5}, h

_{6}}. A is the set of parameters and A = {ε

_{1}, ε

_{2}, ε

_{3}, ε

_{4}, ε

_{5}} = {power, cheap, security, ride comfort, braking performance}. Let (F, A) be an interval-valued fuzzy soft set over the universe U, as shown in Table 1. First of all, according to the above definition of the mean degree of membership, it is easy to compute the average table of (F, A), as shown in Table 5 below.

_{j}(j = 1, 2, 3, 4, 5), for the ε

_{1}, ε

_{2}, ε

_{3}, ε

_{4}and ε

_{5}parameters in each column of the average table. The results are shown in Table 6 below.

_{12}according to the above Definition 5.

_{12}, we can get i = 1, and j = 2.

_{12}is the sum of the non-negative values of the below-mentioned limited list:

_{12}according to the above definition of b

_{12}, which is

_{i}for each object h

_{i}, which scores are illustrated in Table 8.

_{5}> h

_{2}> h

_{6}> h

_{1}> h

_{4}> h

_{3}, and determine that h

_{5}has the highest score value 3.06. Consequently, we decide to choose h

_{5}as the best solution.

## 4. Comparison with the Method

#### 4.1. Using Yang et al.’s Algorithm

_{1}is identical to the score of h

_{2}, namely r

_{1}= r

_{2}= 0. Consequentially, we can determine that both of the objects can be regarded as the best option by means of Yang et al.’s algorithm, which cannot triumphantly address this decision’s difficulty.

#### 4.2. Using Feng et al.’s Algorithm

_{1}is identical to the choice value of h

_{2}, namely c

_{1}= c

_{2}= 3. Therefore, we see that both of the objects can be regarded as the best option according to Feng et al.’s method, which cannot effectively make a decision between the two objects.

#### 4.3. Using Our Proposed Algorithm

_{i}for each object h

_{i}, which scores are illustrated in Table 15.

_{2}has a higher score than h

_{1}. For this reason, we ought to choose h

_{2}as the best option. Therefore, our proposed algorithm is more feasible and efficient, and the results show that our algorithm has a stronger decision-making ability compared to the two existing methods.

## 5. Application of Our Algorithm in a Practical Situation

_{1}, h

_{2}, h

_{3}, …, h

_{19}} = {Xiamen Aishang Inn, That Year Yishe Guest House, Liangzhu Lifestyle Hotel, Xiamen Shibajian Inn, Meng Shi Guang Homestay, Sunny Sea House, Logom Xiamen Moonwatcher Seascape Inn, Xiamen Sunshine Beach Inn, Xiamen Banpo Inn, Fenghuang Mu Coffee Guest House, Xiamen Into Spring Hometel, Garden Dreamer, Xiamen Chenxi Garden, Xia Men Jia No.17, Xiamen Bloom Pinellia Holiday Home, Xiamen Slow Life Hotel, Youran Hotel, Mansion 1929, Seclusion light luxury Guesthouse}, and there is a related set of six parameters, that is, A = {“Environment and cleanliness”, “Position”, “Service”, “Facilities”, “Comfort level”, “Cost performance”}. Table 16 below shows the interval-valued fuzzy soft set (F, A) as a tabular form of the 19 homestays in Siming District, Xiamen.

_{i}for each object h

_{i}, which are illustrated in 19.

_{7}> h

_{12}> h

_{3}> h

_{11}> h

_{4}> h

_{2}> h

_{6}> h

_{5}> h

_{18}> h

_{15}> h

_{1}> h

_{16}> h

_{19}> h

_{17}> h

_{8}> h

_{14}> h

_{9}> h

_{10}> h

_{13}.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} |
---|---|---|---|---|---|

h_{1} | [0.3,0.5] | [0.5,0.6] | [0.4,0.6] | [0.3,0.5] | [0.7,0.9] |

h_{2} | [0.5,0.7] | [0.6,0.7] | [0.6,0.7] | [0.1,0.2] | [0.5,0.8] |

h_{3} | [0.4,0.5] | [0.1,0.3] | [0.4,0.5] | [0.6,0.7] | [0.2,0.4] |

h_{4} | [0.5,0.6] | [0.2,0.3] | [0.7,0.9] | [0.2,0.4] | [0.1,0.3] |

h_{5} | [0.8,1.0] | [0.0,0.2] | [0.7,0.8] | [0.7,0.9] | [0.4,0.6] |

h_{6} | [0.5,0.8] | [0.5,0.7] | [0.5,0.8] | [0.4,0.5] | [0.3,0.4] |

U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} | c_{i} | r_{i} |
---|---|---|---|---|---|---|---|

h_{1} | [0.3,0.5] | [0.5,0.6] | [0.4,0.6] | [0.3,0.5] | [0.7,0.9] | [2.2,3.1] | 1.3 |

h_{2} | [0.5,0.7] | [0.6,0.7] | [0.6,0.7] | [0.1,0.2] | [0.5,0.8] | [2.3,3.1] | 1.9 |

h_{3} | [0.4,0.5] | [0.1,0.3] | [0.4,0.5] | [0.6,0.7] | [0.2,0.4] | [1.7,2.4] | −5.9 |

h_{4} | [0.5,0.6] | [0.2,0.3] | [0.7,0.9] | [0.2,0.4] | [0.1,0.3] | [1.7,2.5] | −5.3 |

h_{5} | [0.8,1.0] | [0.0,0.2] | [0.7,0.8] | [0.7,0.9] | [0.4,0.6] | [2.6,3.5] | 6.1 |

h_{6} | [0.5,0.8] | [0.5,0.7] | [0.5,0.8] | [0.4,0.5] | [0.3,0.4] | [2.2,3.2] | 1.9 |

**Table 3.**Pessimistic reduct fuzzy soft set ${\delta}_{-}=({F}_{-},A)$ of the interval-valued fuzzy soft set (F, A).

U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} |
---|---|---|---|---|---|

h_{1} | 0.3 | 0.5 | 0.4 | 0.3 | 0.7 |

h_{2} | 0.5 | 0.6 | 0.6 | 0.1 | 0.5 |

h_{3} | 0.4 | 0.1 | 0.4 | 0.6 | 0.2 |

h_{4} | 0.5 | 0.2 | 0.7 | 0.2 | 0.1 |

h_{5} | 0.8 | 0.0 | 0.7 | 0.7 | 0.4 |

h_{6} | 0.5 | 0.5 | 0.5 | 0.4 | 0.3 |

U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} | c_{i} |
---|---|---|---|---|---|---|

h_{1} | 0 | 0 | 0 | 0 | 1 | 1 |

h_{2} | 0 | 1 | 0 | 0 | 0 | 1 |

h_{3} | 0 | 0 | 0 | 0 | 0 | 0 |

h_{4} | 0 | 0 | 1 | 0 | 0 | 1 |

h_{5} | 1 | 0 | 1 | 1 | 0 | 3 |

h_{6} | 0 | 0 | 0 | 0 | 0 | 0 |

U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} |
---|---|---|---|---|---|

h_{1} | 0.40 | 0.55 | 0.50 | 0.40 | 0.80 |

h_{2} | 0.60 | 0.65 | 0.65 | 0.15 | 0.65 |

h_{3} | 0.45 | 0.20 | 0.45 | 0.65 | 0.30 |

h_{4} | 0.55 | 0.25 | 0.80 | 0.30 | 0.20 |

h_{5} | 0.90 | 0.10 | 0.75 | 0.80 | 0.50 |

h_{6} | 0.65 | 0.60 | 0.65 | 0.45 | 0.35 |

U | ε_{1} | ε_{2} | ε_{3} | ε_{4} | ε_{5} |
---|---|---|---|---|---|

h_{1} | 0.40 | 0.55 | 0.50 | 0.40 | 0.80 |

h_{2} | 0.60 | 0.65 | 0.65 | 0.15 | 0.65 |

h_{3} | 0.45 | 0.20 | 0.45 | 0.65 | 0.30 |

h_{4} | 0.55 | 0.25 | 0.80 | 0.30 | 0.20 |

h_{5} | 0.90 | 0.10 | 0.75 | 0.80 | 0.50 |

h_{6} | 0.65 | 0.60 | 0.65 | 0.45 | 0.35 |

Q_{j} | 0.90 | 0.65 | 0.80 | 0.80 | 0.80 |

h_{1} | h_{2} | h_{3} | h_{4} | h_{5} | h_{6} | |
---|---|---|---|---|---|---|

h_{1} | 0 | 0.50 | 1.23 | 1.34 | 1.07 | 0.56 |

h_{2} | 0.56 | 0 | 1.55 | 1.23 | 1.03 | 0.45 |

h_{3} | 0.37 | 0.63 | 0 | 0.56 | 0.15 | 0.25 |

h_{4} | 0.54 | 0.38 | 0.63 | 0 | 0.29 | 0.19 |

h_{5} | 1.37 | 1.27 | 1.31 | 1.39 | 0 | 1.03 |

h_{6} | 0.60 | 0.43 | 1.15 | 1.02 | 0.77 | 0 |

Row-Sum M_{i} | Column-Sum N_{i} | Score S_{i} | |
---|---|---|---|

h_{1} | 4.7 | 3.44 | 1.26 |

h_{2} | 4.82 | 3.21 | 1.61 |

h_{3} | 1.96 | 5.87 | −3.91 |

h_{4} | 2.03 | 5.54 | −3.51 |

h_{5} | 6.37 | 3.31 | 3.06 |

h_{6} | 3.97 | 2.48 | 1.49 |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} |
---|---|---|---|---|---|

h_{1} | [0.6,0.8] | [0.4,0.5] | [0.3,0.5] | [0.7,0.9] | [0.1,0.5] |

h_{2} | [0.2,0.7] | [0.3,0.5] | [0.5,0.6] | [0.4,0.8] | [0.6,0.7] |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | c_{i} | r_{i} |
---|---|---|---|---|---|---|---|

h_{1} | [0.6,0.8] | [0.4,0.5] | [0.3,0.5] | [0.7,0.9] | [0.1,0.5] | [2.1,3.2] | 0 |

h_{2} | [0.2,0.7] | [0.3,0.5] | [0.5,0.6] | [0.4,0.8] | [0.6,0.7] | [2.0,3.3] | 0 |

**Table 11.**Optimistic reduct fuzzy soft set ${\xi}_{+}=({Z}_{+},B)$ of the interval-valued fuzzy soft set (Z, B).

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} |
---|---|---|---|---|---|

h_{1} | 0.80 | 0.50 | 0.50 | 0.90 | 0.50 |

h_{2} | 0.70 | 0.50 | 0.60 | 0.80 | 0.70 |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | c_{1} |
---|---|---|---|---|---|---|

h_{1} | 1 | 1 | 0 | 1 | 0 | 3 |

h_{2} | 0 | 1 | 1 | 0 | 1 | 3 |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} |
---|---|---|---|---|---|

h_{1} | 0.70 | 0.45 | 0.40 | 0.80 | 0.30 |

h_{2} | 0.45 | 0.40 | 0.55 | 0.60 | 0.65 |

Q_{j} | 0.70 | 0.45 | 0.55 | 0.80 | 0.65 |

h_{1} | h_{2} | |
---|---|---|

h_{1} | 0 | 0.72 |

h_{2} | 0.81 | 0 |

Row-Sum M_{i} | Column-Sum N_{i} | Score S_{i} | |
---|---|---|---|

h_{1} | 0.72 | 0.81 | −0.09 |

h_{2} | 0.81 | 0.72 | 0.09 |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | e_{6} |
---|---|---|---|---|---|---|

h_{1} | [0.72, 0.88] | [0.78, 0.89] | [0.84, 0.96] | [0.79, 0.88] | [0.75, 0.85] | [0.75, 0.85] |

h_{2} | [0.78, 0.91] | [0.88, 0.89] | [0.93, 0.97] | [0.80, 0.92] | [0.78, 0.89] | [0.77, 0.86] |

h_{3} | [0.90, 0.93] | [0.89, 0.93] | [0.95, 0.98] | [0.85, 0.92] | [0.88, 0.92] | [0.84, 0.89] |

h_{4} | [0.85, 0.95] | [0.83, 0.95] | [0.84, 0.96] | [0.81, 0.94] | [0.82, 0.93] | [0.78, 0.94] |

h_{5} | [0.67, 1.00] | [0.60, 1.00] | [0.85, 1.00] | [0.70, 1.00] | [0.70, 1.00] | [0.70, 1.00] |

h_{6} | [0.79, 1.00] | [0.75, 0.88] | [0.85, 1.00] | [0.83, 1.00] | [0.79, 0.88] | [0.75, 0.85] |

h_{7} | [0.89, 1.00] | [0.85, 0.94] | [0.93, 1.00] | [0.96, 1.00] | [0.89, 1.00] | [0.89, 0.97] |

h_{8} | [0.67, 0.96] | [0.58, 0.96] | [0.92, 1.00] | [0.50, 0.95] | [0.58, 0.96] | [0.58, 0.96] |

h_{9} | [0.54, 0.83] | [0.58, 0.84] | [0.67, 0.94] | [0.58, 0.83] | [0.54, 0.85] | [0.58, 0.83] |

h_{10} | [0.25, 1.00] | [0.75, 1.00] | [0.50, 1.00] | [0.50, 0.96] | [0.25, 1.00] | [0.25, 0.92] |

h_{11} | [0.84, 0.92] | [0.86, 0.90] | [0.89, 0.99] | [0.89, 0.92] | [0.86, 0.93] | [0.85, 0.89] |

h_{12} | [0.84, 1.00] | [0.79, 1.00] | [0.81, 1.00] | [0.84, 1.00] | [0.81, 1.00] | [0.81, 1.00] |

h_{13} | [0.38, 0.75] | [0.38, 0.75] | [0.75, 0.75] | [0.50, 0.75] | [0.50, 0.75] | [0.38, 0.75] |

h_{14} | [0.67, 0.80] | [0.83, 0.94] | [0.67, 0.85] | [0.67, 0.80] | [0.67, 0.75] | [0.67, 0.80] |

h_{15} | [0.75, 0.88] | [0.80, 0.95] | [0.81, 1.00] | [0.74, 1.00] | [0.72, 0.92] | [0.68, 0.88] |

h_{16} | [0.73, 0.93] | [0.83, 0.96] | [0.79, 0.96] | [0.67, 0.93] | [0.65, 0.89] | [0.65, 0.93] |

h_{17} | [0.75, 0.89] | [0.83, 0.91] | [0.80, 0.89] | [0.73, 0.85] | [0.70, 0.83] | [0.68, 0.83] |

h_{18} | [0.71, 0.89] | [0.86, 0.93] | [0.88, 0.96] | [0.79, 0.93] | [0.75, 0.89] | [0.67, 0.88] |

h_{19} | [0.70, 0.92] | [0.78, 0.83] | [0.80, 0.92] | [0.78, 0.92] | [0.73, 0.85] | [0.65, 0.84] |

U | e_{1} | e_{2} | e_{3} | e_{4} | e_{5} | e_{6} |
---|---|---|---|---|---|---|

h_{1} | 0.80 | 0.84 | 0.90 | 0.84 | 0.80 | 0.80 |

h_{2} | 0.85 | 0.89 | 0.95 | 0.86 | 0.84 | 0.82 |

h_{3} | 0.92 | 0.91 | 0.97 | 0.89 | 0.90 | 0.87 |

h_{4} | 0.90 | 0.89 | 0.90 | 0.88 | 0.88 | 0.86 |

h_{5} | 0.84 | 0.80 | 0.93 | 0.85 | 0.85 | 0.85 |

h_{6} | 0.90 | 0.82 | 0.93 | 0.92 | 0.84 | 0.80 |

h_{7} | 0.95 | 0.90 | 0.97 | 0.98 | 0.95 | 0.93 |

h_{8} | 0.82 | 0.77 | 0.96 | 0.73 | 0.77 | 0.77 |

h_{9} | 0.69 | 0.71 | 0.81 | 0.71 | 0.70 | 0.71 |

h_{10} | 0.63 | 0.88 | 0.75 | 0.73 | 0.63 | 0.59 |

h_{11} | 0.88 | 0.88 | 0.94 | 0.91 | 0.90 | 0.87 |

h_{12} | 0.92 | 0.90 | 0.91 | 0.92 | 0.91 | 0.91 |

h_{13} | 0.57 | 0.57 | 0.75 | 0.63 | 0.63 | 0.57 |

h_{14} | 0.74 | 0.89 | 0.76 | 0.74 | 0.71 | 0.74 |

h_{15} | 0.82 | 0.88 | 0.91 | 0.87 | 0.82 | 0.78 |

h_{16} | 0.83 | 0.90 | 0.88 | 0.80 | 0.77 | 0.79 |

h_{17} | 0.82 | 0.87 | 0.85 | 0.79 | 0.77 | 0.76 |

h_{18} | 0.80 | 0.90 | 0.92 | 0.86 | 0.82 | 0.78 |

h_{19} | 0.81 | 0.81 | 0.86 | 0.85 | 0.79 | 0.75 |

Q_{j} | 0.95 | 0.91 | 0.97 | 0.98 | 0.95 | 0.93 |

U | h_{1} | h_{2} | h_{3} | h_{4} | h_{5} | h_{6} | h_{7} | h_{8} | h_{9} | h_{10} | h_{11} | h_{12} | h_{13} | h_{14} | h_{15} | h_{16} | h_{17} | h_{18} | h_{19} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

h_{1} | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.02 | 0.00 | 0.25 | 0.69 | 0.85 | 0.00 | 0.00 | 1.33 | 0.47 | 0.02 | 0.10 | 0.18 | 0.02 | 0.14 |

h_{2} | 0.24 | 0.00 | 0.00 | 0.05 | 0.14 | 0.12 | 0.00 | 0.42 | 0.93 | 1.05 | 0.02 | 0.04 | 1.58 | 0.66 | 0.15 | 0.26 | 0.37 | 0.15 | 0.36 |

h_{3} | 0.51 | 0.26 | 0.00 | 0.16 | 0.36 | 0.30 | 0.01 | 0.68 | 1.19 | 1.31 | 0.11 | 0.07 | 1.84 | 0.92 | 0.40 | 0.51 | 0.63 | 0.40 | 0.62 |

h_{4} | 0.35 | 0.16 | 0.00 | 0.00 | 0.24 | 0.18 | 0.00 | 0.58 | 1.04 | 1.16 | 0.03 | 0.00 | 1.68 | 0.76 | 0.25 | 0.37 | 0.47 | 0.27 | 0.47 |

h_{5} | 0.19 | 0.04 | 0.00 | 0.03 | 0.00 | 0.06 | 0.00 | 0.35 | 0.83 | 1.04 | 0.00 | 0.02 | 1.48 | 0.66 | 0.15 | 0.26 | 0.35 | 0.16 | 0.27 |

h_{6} | 0.26 | 0.11 | 0.03 | 0.07 | 0.16 | 0.00 | 0.00 | 0.44 | 0.92 | 1.11 | 0.03 | 0.02 | 1.57 | 0.73 | 0.20 | 0.33 | 0.42 | 0.22 | 0.36 |

h_{7} | 0.74 | 0.49 | 0.24 | 0.39 | 0.59 | 0.50 | 0.00 | 0.91 | 1.42 | 1.54 | 0.32 | 0.22 | 2.07 | 1.15 | 0.63 | 0.74 | 0.86 | 0.63 | 0.85 |

h_{8} | 0.08 | 0.01 | 0.00 | 0.06 | 0.03 | 0.03 | 0.00 | 0.00 | 0.52 | 0.76 | 0.02 | 0.05 | 1.16 | 0.39 | 0.05 | 0.08 | 0.12 | 0.06 | 0.14 |

h_{9} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.33 | 0.00 | 0.00 | 0.65 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

h_{10} | 0.04 | 0.00 | 0.00 | 0.00 | 0.09 | 0.07 | 0.00 | 0.12 | 0.21 | 0.00 | 0.00 | 0.00 | 0.53 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.08 |

h_{11} | 0.42 | 0.20 | 0.02 | 0.10 | 0.28 | 0.21 | 0.00 | 0.61 | 1.11 | 1.23 | 0.00 | 0.03 | 1.76 | 0.85 | 0.32 | 0.45 | 0.54 | 0.34 | 0.54 |

h_{12} | 0.52 | 0.32 | 0.08 | 0.17 | 0.39 | 0.30 | 0.00 | 0.74 | 1.20 | 1.32 | 0.13 | 0.00 | 1.85 | 0.93 | 0.41 | 0.52 | 0.64 | 0.42 | 0.64 |

h_{13} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

h_{14} | 0.05 | 0.00 | 0.00 | 0.00 | 0.10 | 0.08 | 0.00 | 0.14 | 0.32 | 0.39 | 0.01 | 0.00 | 0.92 | 0.00 | 0.01 | 0.00 | 0.02 | 0.00 | 0.09 |

h_{15} | 0.13 | 0.01 | 0.00 | 0.01 | 0.11 | 0.07 | 0.00 | 0.33 | 0.79 | 0.91 | 0.00 | 0.00 | 1.44 | 0.53 | 0.00 | 0.15 | 0.23 | 0.03 | 0.22 |

h_{16} | 0.10 | 0.01 | 0.00 | 0.01 | 0.11 | 0.09 | 0.00 | 0.25 | 0.68 | 0.80 | 0.02 | 0.00 | 1.33 | 0.41 | 0.04 | 0.00 | 0.12 | 0.04 | 0.18 |

h_{17} | 0.05 | 0.00 | 0.00 | 0.00 | 0.08 | 0.05 | 0.00 | 0.17 | 0.56 | 0.69 | 0.00 | 0.00 | 1.21 | 0.31 | 0.00 | 0.00 | 0.00 | 0.02 | 0.09 |

h_{18} | 0.13 | 0.01 | 0.00 | 0.03 | 0.12 | 0.09 | 0.00 | 0.34 | 0.79 | 0.91 | 0.02 | 0.01 | 1.44 | 0.52 | 0.03 | 0.16 | 0.25 | 0.00 | 0.23 |

h_{19} | 0.02 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.19 | 0.57 | 0.77 | 0.00 | 0.00 | 1.22 | 0.38 | 0.00 | 0.07 | 0.09 | 0.01 | 0.00 |

U | Row-Sum M_{i} | Column-Sum N_{i} | Score S_{i} |
---|---|---|---|

h_{1} | 4.11 | 3.83 | 0.28 |

h_{2} | 6.54 | 1.62 | 4.92 |

h_{3} | 10.28 | 0.37 | 9.91 |

h_{4} | 8.01 | 1.08 | 6.93 |

h_{5} | 5.89 | 2.85 | 3.04 |

h_{6} | 6.98 | 2.17 | 4.81 |

h_{7} | 14.29 | 0.01 | 14.28 |

h_{8} | 3.56 | 6.52 | −2.96 |

h_{9} | 1.03 | 13.77 | −12.74 |

h_{10} | 1.15 | 16.17 | −15.02 |

h_{11} | 9.01 | 0.71 | 8.30 |

h_{12} | 10.58 | 0.46 | 10.12 |

h_{13} | 0.00 | 25.06 | −25.06 |

h_{14} | 2.13 | 9.72 | −7.59 |

h_{15} | 4.96 | 2.66 | 2.30 |

h_{16} | 4.19 | 4.00 | 0.19 |

h_{17} | 3.23 | 5.30 | −2.07 |

h_{18} | 5.08 | 2.77 | 2.31 |

h_{19} | 3.33 | 5.28 | −1.95 |

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## Share and Cite

**MDPI and ACS Style**

Ma, X.; Wang, Y.; Qin, H.; Wang, J.
A Decision-Making Algorithm Based on the Average Table and Antitheses Table for Interval-Valued Fuzzy Soft Set. *Symmetry* **2020**, *12*, 1131.
https://doi.org/10.3390/sym12071131

**AMA Style**

Ma X, Wang Y, Qin H, Wang J.
A Decision-Making Algorithm Based on the Average Table and Antitheses Table for Interval-Valued Fuzzy Soft Set. *Symmetry*. 2020; 12(7):1131.
https://doi.org/10.3390/sym12071131

**Chicago/Turabian Style**

Ma, Xiuqin, Yanan Wang, Hongwu Qin, and Jin Wang.
2020. "A Decision-Making Algorithm Based on the Average Table and Antitheses Table for Interval-Valued Fuzzy Soft Set" *Symmetry* 12, no. 7: 1131.
https://doi.org/10.3390/sym12071131