# Measuring Product Similarity with Hesitant Fuzzy Set for Recommendation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Feature Description of Sparse Matrix in the E-commerce Recommendation System

## 3. The Introduction of a Hesitant Fuzzy Set

#### 3.1. Basic Definition

**Definition**

**1**

**[29].**Let X be a reference set. Hesitant fuzzy set (HFS) A is a set of different numbers of membership functions ${h}_{A}(x)$ on X valued on [0, 1].

**Example**

**1**

**[28]**. Let $X=\{{x}_{1},{x}_{2},{x}_{3}\}$ be a fixed set, ${h}_{A}({x}_{1})=\{0.2,0.4,0.5\}$, ${h}_{A}({x}_{2})=\{0.3,0.4\}$ and ${h}_{A}({x}_{3})=\{0.3,0.2,0.5,0.6\}$ be the HFEs of ${x}_{i}(i=1,2,3)$ to a set $A$ respectively. Then $A$ can be considered as a HFS:

#### 3.2. Similarity

**Definition**

**2**

**[32].**Let${A}_{1}$and${A}_{2}$be two HFSs on$X$, then the distance between ${A}_{1}$and${A}_{2}$is defined as$d({A}_{1},{A}_{2})$, which satisfies the following properties:

- (1)
- $0\le d({A}_{1},{A}_{2})\le 1$.
- (2)
- $d({A}_{1},{A}_{2})=0$ if and only if ${A}_{1}={A}_{2}$.
- (3)
- $d({A}_{1},{A}_{2})=d({A}_{2},{A}_{1})$.

**Definition**

**3**

- (1)
- $0\le S({A}_{1},{A}_{2})\le 1$.
- (2)
- $S({A}_{1},{A}_{2})=1$ if and only if ${A}_{1}={A}_{2}$.
- (3)
- $S({A}_{1},{A}_{2})=S({A}_{2},{A}_{1})$.

_{1}and h

_{2}, based on the distance measurement formula:

## 4. Construction of a Similarity Model of Sparse Matrix Products

#### 4.1. Affiliation of User Ratings

#### 4.2. Product Rating Representation

#### 4.3. Horizontal Comparison of Products

- (1)
- Forward strategy: If the average of all ratings ${R}_{ij}(j=1,2,\cdots )$ of historical products from $Use{r}_{i}$, which is expressed as $\overline{{R}_{i}}$, meet $\overline{{R}_{i}}\le 4$, indicating that $Use{r}_{i}$ is a pessimistic user, $(l-{l}_{j})$${\gamma}_{{l}_{j}j}$ is added before the first element of $h(Ite{m}_{j})$. Hence, ${\gamma}_{1j}={\gamma}_{2j}=\cdots ={\gamma}_{l-{l}_{j},j}={\gamma}_{1j}$.
- (2)
- Backward strategy: If the average of all ratings ${R}_{ij}(j=1,2,\cdots )$ of historical products from $Use{r}_{i}$, which is expressed as $\overline{{R}_{i}}$, meet $\overline{{R}_{i}}>4$, indicating that $Use{r}_{i}$ is an optimistic user, $(l-{l}_{j})$${\gamma}_{{l}_{j}j}$ is added after the last element of $h(Ite{m}_{j})$. Hence, ${\gamma}_{{l}_{j}+1,j}={\gamma}_{{l}_{j}+2,j}=\cdots ={\gamma}_{l,j}={\gamma}_{{l}_{j},j}$.

#### 4.4. Similarity Calculation of Products

#### 4.5. Algorithm Implementation of Product Recommendation

## 5. Case Application

## 6. Algorithm Verification

## 7. Conclusions and Prospect

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Hesitant Fuzzy Set | Recommendation System | |
---|---|---|

Concept description | Describe a thing with a set of membership degrees | Describe a product with a set of history ratings |

Characteristic | Use a set of numbers to express the degree of uncertainty of the object to be evaluated | Use a set of historical ratings to express the uncertainty of the user’s attitude |

Data form | The difference in membership stems from the different attitudes of the raters | The difference in ratings stems from the user’s satisfaction with the product |

Key element | Membership functions | User ratings |

Numerical comparison | The degree of membership can be compared; | User ratings can be compared horizontally; |

the same degree of membership means the same attitude of users | the same user rating represents the same level of user satisfaction |

Movie ID | 260 | 293 | 316 | 349 | 457 | 527 | 647 | 736 | 1222 | 2502 |
---|---|---|---|---|---|---|---|---|---|---|

User_{1} | 5.0 | 3.0 | 4.0 | 5.0 | 5.0 | 3.0 | 5.0 | 5.0 | ||

User_{6} | 3.0 | 5.0 | 5.0 | 5.0 | 3.0 | 4.0 | 5.0 | |||

User_{17} | 5.0 | 3.5 | 4.5 | 4.5 | 4.5 | |||||

User_{28} | 4.0 | 4.5 | 4.0 | 3.0 | 3.0 | 3.0 | 2.5 | 2.5 | 4.0 | 2.0 |

User_{42} | 5.0 | 2.0 | 4.0 | 4.0 | 4.0 | 2.0 | 5.0 | 5.0 | ||

User_{57} | 5.0 | 3.0 | 4.0 | 5.0 | 5.0 | 2.0 | 4.0 | |||

User_{64} | 3.5 | 4.0 | 3.0 | 5.0 | 3.5 | 3.5 | 4.0 | 4.5 | ||

User_{68} | 5.0 | 4.0 | 3.5 | 4.5 | 4.0 | 3.5 | 4.0 | 2.5 | 5.0 | |

User_{84} | 4.0 | 4.0 | 4.0 | 5.0 | 4.0 | 3.0 | ||||

User_{91} | 4.5 | 4.0 | 4.0 | 3.0 | 4.0 | 2.5 | 4.5 |

Movie ID | 260 | 293 | 316 | 349 | 457 | 527 | 647 | 736 | 1222 | 2502 |
---|---|---|---|---|---|---|---|---|---|---|

User_{1} | 1 | 0.6 | 0.8 | 1 | 1 | 0.6 | 1 | 1 | ||

User_{6} | 0.6 | 1 | 1 | 1 | 0.6 | 0.8 | 1 | |||

User_{17} | 1 | 0.7 | 0.9 | 0.9 | 0.9 | |||||

User_{28} | 0.8 | 0.9 | 0.8 | 0.6 | 0.6 | 0.6 | 0.5 | 0.5 | 0.8 | 0.4 |

User_{42} | 1 | 0.4 | 0.8 | 0.8 | 0.8 | 0.4 | 1 | 1 | ||

User_{57} | 1 | 0.6 | 0.8 | 1 | 1 | 0.4 | 0.8 | |||

User_{64} | 0.7 | 0.8 | 0.6 | 1 | 0.7 | 0.7 | 0.8 | 0.9 | ||

User_{68} | 1 | 0.8 | 0.7 | 0.9 | 0.8 | 0.7 | 0.8 | 0.5 | 1 | |

User_{84} | 0.8 | 0.8 | 0.8 | 1 | 0.8 | 0.6 | ||||

User_{91} | 0.9 | 0.8 | 0.8 | 0.6 | 0.8 | 0.5 | 0.9 |

Movie ID | 260 | 293 | 316 | 349 | 457 | 527 | 647 | 736 | 1222 | 2502 |
---|---|---|---|---|---|---|---|---|---|---|

260 | 1 | 0.996080 | 0.981830 | 0.994104 | 0.995393 | 0.996563 | 0.978223 | 0.799065 | 0.988359 | 0.975629 |

293 | 1 | 0.989067 | 0.998548 | 0.995909 | 0.995856 | 0.989216 | 0.986584 | 0.989434 | 0.973720 | |

316 | 1 | 0.989887 | 0.993023 | 0.990814 | 0.990538 | 0.996927 | 0.999930 | 0.980996 | ||

349 | 1 | 0.993149 | 0.993961 | 0.987747 | 0.989801 | 0.989588 | 0.974983 | |||

457 | 1 | 0.998860 | 0.987245 | 0.986571 | 0.992389 | 0.979900 | ||||

527 | 1 | 0.986705 | 0.984144 | 0.991993 | 0.953648 | |||||

647 | 1 | 0.991155 | 0.989228 | 0.978055 | ||||||

736 | 1 | 0.990056 | 0.976978 | |||||||

1222 | 1 | 0.994563 | ||||||||

2502 | 1 |

User_{i} | 1 | 6 | 17 | 28 | 42 | 57 | 64 | 68 | 84 | 91 |
---|---|---|---|---|---|---|---|---|---|---|

User_{58} | 0.4 | 0.5 | 0.2 | 0.5 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 |

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**MDPI and ACS Style**

Cui, C.; Li, J.; Zang, Z.
Measuring Product Similarity with Hesitant Fuzzy Set for Recommendation. *Mathematics* **2021**, *9*, 2657.
https://doi.org/10.3390/math9212657

**AMA Style**

Cui C, Li J, Zang Z.
Measuring Product Similarity with Hesitant Fuzzy Set for Recommendation. *Mathematics*. 2021; 9(21):2657.
https://doi.org/10.3390/math9212657

**Chicago/Turabian Style**

Cui, Chunsheng, Jielu Li, and Zhenchun Zang.
2021. "Measuring Product Similarity with Hesitant Fuzzy Set for Recommendation" *Mathematics* 9, no. 21: 2657.
https://doi.org/10.3390/math9212657