# A Novel Extension of the TOPSIS Method Adapted for the Use of Single-Valued Neutrosophic Sets and Hamming Distance for E-Commerce Development Strategies Selection

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

_{A}, I

_{A}, and F

_{A}$\in \text{}{[}^{-}0,{1}^{+}]$and satisfy the following constraint$0\le {\mu}_{A}(x)+{\pi}_{A}(x)+{\nu}_{A}(x)\le 3$.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

_{1}, x

_{2}, ..., x

_{n}) and Y = (y

_{1}, y

_{2}, ..., y

_{n}) be two n-dimensional vectors,${x}_{i}=<{t}_{xi},{i}_{xi},{f}_{xi}>$and${y}_{i}=<{t}_{yi},{i}_{yi},{f}_{yi}>$. The Hamming distance between X and Y is defined as

**Definition**

**6.**

_{1}, x

_{2}, ..., x

_{n}) and Y = (y

_{1}, y

_{2}, ..., y

_{n}) be two n-dimensional vectors,${x}_{i}=<{t}_{xi},{i}_{xi},{f}_{xi}>$and${y}_{i}=<{t}_{yi},{i}_{yi},{f}_{yi}>$. The Euclidean distance between X and Y is defined as

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

_{j}is

_{j}is the element j of the weighting vector,${w}_{j}\in [0,\text{}1]$and${\sum}_{j=1}^{n}{w}_{j}}=1$.

## 3. The TOPSIS Method Customized to the Use of SVNNs and Group Decision-Making

#### 3.1. The TOPSIS Method

_{i}of the ith alternative to the ideal and anti-ideal solutions is calculated as

#### 3.2. An Extension of the TOPSIS Method Adapted for the Use of SVNNs

_{ij}in the evaluation matrix D means the rating of the alternative i with respect to the criterion j and entries w

_{j}in W of the weight vector denote the weights of the criterion j, for each i = 1, … m and j = 1, ..., n.

## 4. A Numerical Illustration

_{j}= 0.20.

## 5. Discussion and Comparison Analysis

_{2}and W

_{5}. In the first case (W

_{2}), both distances gave the same ranking order, whereas in the second case (W

_{5}), there was a difference in the second- and third-ranked alternatives.

_{j}= 0.20. The achieved ranking results are shown in Table 13.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Alternatives | Designation |
---|---|

A_{1}—E-customization and personalization—Ansari & Mela [70] | ECDS1 |

A_{2}—Social E-commerce adoption model—Hajli [71] | ECDS2 |

A_{3}—Strong search engine optimization (SEO)—Sen [72] | ECDS3 |

Criteria | Designation |
---|---|

C_{1}—Feasibility of the strategy | FS |

C_{2}—Implementation speed | IS |

C_{3}—Compliance with the corporate strategy | CS |

C_{4}—Compliance of the strategy with the mission and vision of the company | MV |

C_{5}—General acceptance | GA |

FS | IS | CS | MV | GA | |
---|---|---|---|---|---|

ECDS1 | <0.6, 0.1, 0.1> | <0.6, 0.1, 0.1> | <0.6, 0.1, 0.1> | <0.4, 0.1, 0.1> | <0.4, 0.1, 0.1> |

ECDS2 | <1.0, 0.0, 0.0> | <0.8, 0.0, 0.0> | <1.0, 0.1, 0.1> | <1.0, 0.1, 0.3> | <1.0, 0.0, 0.1> |

ECDS3 | <0.6, 0.0, 0.2> | <0.6, 0.2, 0.1> | <0.8, 0.2, 0.1> | <1.0, 0.2, 0.3> | <1.0, 0.0, 0.2> |

FS | IS | CS | MV | GA | |
---|---|---|---|---|---|

ECDS1 | <0.5, 0.0, 0.1> | <0.7, 0.1, 0.1> | <0.5, 0.0, 0.1> | <0.4, 0.1, 0.1> | <0.4, 0.0, 0.1> |

ECDS2 | <0.9, 0.0, 0.0> | <0.7, 0.1, 0.0> | <0.9, 0.0, 0.0> | <1.0, 0.0, 0.1> | <0.7, 0.0, 0.2> |

ECDS3 | <0.7, 0.0, 0.0> | <0.6, 0.1, 0.1> | <0.8, 0.1, 0.2> | <0.9, 0.1, 0.3> | <0.8, 0.0, 0.2> |

FS | IS | CS | MV | GA | |
---|---|---|---|---|---|

ECDS1 | <0.5, 0.0, 0.0> | <0.8, 0.1, 0.1> | <0.6, 0.0, 0.1> | <0.5, 0.0, 0.0> | <0.5, 0.1, 0.1> |

ECDS2 | <0.8, 0.0, 0.1> | <0.7, 0.0, 0.0> | <1.0, 0.0, 0.0> | <0.9, 0.0, 0.1> | <0.6, 0.0, 0.1> |

ECDS3 | <0.8, 0.1, 0.1> | <0.7, 0.0, 0.0> | <0.8, 0.0, 0.1> | <0.9, 0.1, 0.2> | <0.8, 0.0, 0.0> |

FS | IS | CS | MV | GA | |
---|---|---|---|---|---|

ECDS1 | <0.5, 0.0, 0.0> | <0.7, 0.1, 0.1> | <0.6, 0.0, 0.1> | <0.4, 0.0, 0.0> | <0.4, 0.0, 0.1> |

ECDS2 | <1.0, 0.0, 0.0> | <0.7, 0.0, 0.0> | <1.0, 0.0, 0.0> | <1.0, 0.0, 0.1> | <1.0, 0.0, 0.1> |

ECDS3 | <0.7, 0.0, 0.0> | <0.6, 0.0, 0.0> | <0.8, 0.0, 0.1> | <1.0, 0.1, 0.3> | <1.0, 0.0, 0.0> |

FS | IS | CS | MV | GA | |
---|---|---|---|---|---|

ECDS^{+} | <1.0, 0.0, 0.0> | <0.7, 0.0, 0.0> | <1.0, 0.0, 0.0> | <1.0, 0.0, 0.0> | <1.0, 0.0, 0.0> |

ECDS^{−} | <0.5, 0.0, 0.0> | <0.6, 0.1, 0.1> | <0.6, 0.0, 0.1> | <0.4, 0.1, 0.3> | <0.4, 0.0, 0.1> |

${\mathit{d}}_{\mathit{i}}^{+}$ | ${\mathit{d}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Rank | |
---|---|---|---|---|

ECDS1 | 0.87 | 0.69 | 0.44 | 3 |

ECDS2 | 0.38 | 1.08 | 0.74 | 1 |

ECDS3 | 0.63 | 0.66 | 0.51 | 2 |

${\mathit{d}}_{\mathit{i}}^{+}$ | ${\mathit{d}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Rank | |
---|---|---|---|---|

ECDS1 | 3.38 | 3.24 | 0.490 | 3 |

ECDS2 | 1.70 | 4.25 | 0.714 | 1 |

ECDS3 | 2.60 | 2.54 | 0.495 | 2 |

w_{1} | w_{2} | w_{3} | w_{4} | w_{5} | Σw_{j} | |
---|---|---|---|---|---|---|

W_{1} | 0.40 | 0.15 | 0.15 | 0.15 | 0.15 | 1.00 |

W_{2} | 0.15 | 0.40 | 0.15 | 0.15 | 0.15 | 1.00 |

W_{3} | 0.15 | 0.15 | 0.40 | 0.15 | 0.15 | 1.00 |

W_{4} | 0.15 | 0.15 | 0.15 | 0.40 | 0.15 | 1.00 |

W_{5} | 0.15 | 0.15 | 0.15 | 0.15 | 0.40 | 1.00 |

W_{1} | W_{2} | W_{3} | W_{4} | W_{5} | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | |

ECDS1 | 0.45 | 3 | 0.51 | 2 | 0.44 | 3 | 0.39 | 3 | 0.43 | 3 |

ECDS2 | 0.74 | 1 | 0.70 | 1 | 0.72 | 1 | 0.75 | 1 | 0.78 | 1 |

ECDS3 | 0.50 | 2 | 0.43 | 3 | 0.54 | 2 | 0.59 | 2 | 0.49 | 2 |

W_{1} | W_{2} | W_{3} | W_{4} | W_{5} | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | ${\mathit{C}}_{\mathit{i}}$ | Rank | |

ECDS1 | 0.49 | 3 | 0.56 | 2 | 0.48 | 3 | 0.44 | 3 | 0.48 | 2 |

ECDS2 | 0.72 | 1 | 0.67 | 1 | 0.70 | 1 | 0.73 | 1 | 0.76 | 1 |

ECDS3 | 0.50 | 2 | 0.41 | 3 | 0.54 | 2 | 0.57 | 2 | 0.46 | 3 |

Overall Ratings | Score | Rank | Cosine | Rank | |
---|---|---|---|---|---|

ECDS1 | <0.55, 0.00, 0.00> | 0.78 | 3 | 0.55 | 3 |

ECDS2 | <1.00, 0.00, 0.00> | 1.00 | 1 | 1.00 | 1 |

ECDS3 | <1.00, 0.00, 0.00> | 1.00 | 1 | 1.00 | 1 |

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**MDPI and ACS Style**

Karabašević, D.; Stanujkić, D.; Zavadskas, E.K.; Stanimirović, P.; Popović, G.; Predić, B.; Ulutaş, A.
A Novel Extension of the TOPSIS Method Adapted for the Use of Single-Valued Neutrosophic Sets and Hamming Distance for E-Commerce Development Strategies Selection. *Symmetry* **2020**, *12*, 1263.
https://doi.org/10.3390/sym12081263

**AMA Style**

Karabašević D, Stanujkić D, Zavadskas EK, Stanimirović P, Popović G, Predić B, Ulutaş A.
A Novel Extension of the TOPSIS Method Adapted for the Use of Single-Valued Neutrosophic Sets and Hamming Distance for E-Commerce Development Strategies Selection. *Symmetry*. 2020; 12(8):1263.
https://doi.org/10.3390/sym12081263

**Chicago/Turabian Style**

Karabašević, Darjan, Dragiša Stanujkić, Edmundas Kazimieras Zavadskas, Predrag Stanimirović, Gabrijela Popović, Bratislav Predić, and Alptekin Ulutaş.
2020. "A Novel Extension of the TOPSIS Method Adapted for the Use of Single-Valued Neutrosophic Sets and Hamming Distance for E-Commerce Development Strategies Selection" *Symmetry* 12, no. 8: 1263.
https://doi.org/10.3390/sym12081263