# A Hybrid Neutrosophic Group ANP-TOPSIS Framework for Supplier Selection Problems

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## Abstract

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## 1. Introduction

#### Research Contribution

- The sustainable supplier selection is a multi-criteria decision-making issue including many conflicting criteria. The valuation and selection of sustainable suppliers is a difficult problem due to vague, inconsistent and imprecise knowledge of decision makers. The literature on supply chain management for measuring green performance, the requirement for methodological analysis of how sustainable variables affect each other and of how to consider vague, imprecise and inconsistent knowledge is somehow inconclusive, but these drawbacks have been treated in our research.
- In most cases, the truth, falsity and indeterminacy degrees cannot be defined precisely in the real selection of sustainable suppliers, but denoted by several possible interval values. Therefore, we presented ANP TOPSIS, and combined them with interval-valued neutrosophic sets to select sustainable suppliers for the first time.
- The integrated framework leads to accurate decisions due to the way it treats uncertainty. The sustainable criteria for selecting suppliers are determined from the cited literature and the features of organizations under analysis. Then, the decision makers gather data and information.
- We select ANP and TOPSIS for solving sustainable supplier selection problems for the following reasons:
- -
- Since the independent concept of criteria is not constantly right and in actual life, there exist criteria dependent on each other, and we used ANP for precise weighting of criteria.
- -
- The ANP needs many pairwise comparison matrices based on numerals and interdependence of criteria and alternatives, and, to escape this drawback, the TOPSIS was used to rank alternatives.
- -
- The main problem of sustainable supplier selection problems is how to design and implement a flexible model for evaluating all available suppliers; since it considers the uncertainty that usually exists in real life, our model is the best.
- -
- The proposed framework is used to study the case of a dairy and foodstuff company in Egypt, and can be employed to solve any sustainable supplier selection problem of any other company.
- -
- Comparison with other existing methods, which are popular and attractive, was presented to validate our model.

## 2. Literature Review

- Cost,
- Quality,
- Flexibility,
- Technology capability.

- Defilement production,
- Resource exhaustion,
- Eco-design and environmental administration.

## 3. Preliminaries

#### 3.1. Interval-Valued Neutrosophic Sets (INS)

#### 3.2. The Related Operations of Interval-Valued Neutrosophic Sets

- AdditionLet ${A}_{1},{A}_{2}$ be two INSs, where${A}_{1}=<\left[{T}_{{A}_{1}}^{L},{T}_{{A}_{1}}^{U}\right],\left[{I}_{{A}_{1}}^{L},{I}_{{A}_{1}}^{U}\right],\left[{F}_{{A}_{1}}^{L},{F}_{{A}_{1}}^{U}\right]>$, ${A}_{2}=<\left[{T}_{{A}_{2}}^{L},{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}\right],\left[{F}_{{A}_{2}}^{L},{F}_{{A}_{2}}^{U}\right]>$ then ${A}_{1}+{A}_{2}=<\left[{T}_{{A}_{1}}^{L}+{T}_{{A}_{2}}^{L}-{T}_{{A}_{1}}^{L}{T}_{{A}_{2}}^{L},{T}_{{A}_{1}}^{U}+{T}_{{A}_{2}}^{U}-{T}_{{A}_{1}}^{U}{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{1}}^{L}{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}{I}_{{A}_{2}}^{U}\right],\left[{F}_{{A}_{1}}^{L}{F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}{F}_{{A}_{2}}^{U}\right]>.$
- Subset${A}_{1}\subseteq {A}_{2}$ if and only if ${T}_{{A}_{1}}^{L}\le {T}_{{A}_{2}}^{L}$,${T}_{{A}_{1}}^{U}\le {T}_{{A}_{2}}^{U}$; ${I}_{{A}_{1}}^{L}\ge {I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}\ge {I}_{{A}_{2}}^{U}$;${F}_{{A}_{1}}^{L}\ge {F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}\ge {F}_{{A}_{2}}^{U}$.
- Equality${A}_{1}={A}_{2}$ if and only if ${A}_{1}\subseteq {A}_{2}$ and ${A}_{2}\subseteq {A}_{1}$.
- ComplementLet $V=<\left[{T}_{V}^{L}\left(x\right),{T}_{V}^{U}\left(x\right)\right],\left[{I}_{V}^{L}\left(x\right),{I}_{V}^{U}\left(x\right)\right],\left[{F}_{V}^{L}\left(x\right),{F}_{V}^{U}\left(x\right)\right]>$, then${V}^{c}$ = $<\left[{F}_{V}^{L}\left(x\right),{F}_{V}^{U}\left(x\right)\right],\left[1-{I}_{V}^{U}\left(x\right),1-{I}_{V}^{L}\left(x\right)\right],\left[{T}_{V}^{L}\left(x\right),{T}_{V}^{U}\left(x\right)\right]>$.
- Multiplication${A}_{1}\times {A}_{2}=<\left[{T}_{{A}_{1}}^{L}{T}_{{A}_{2}}^{L},{T}_{{A}_{1}}^{U}{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{1}}^{L}+{I}_{{A}_{2}}^{L}-{I}_{{A}_{1}}^{L}{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}+{I}_{{A}_{2}}^{U}-{I}_{{A}_{1}}^{U}{I}_{{A}_{2}}^{U}\right],$$\left[{F}_{{A}_{1}}^{L}+{F}_{{A}_{2}}^{L}-{F}_{{A}_{1}}^{L}{F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}+{F}_{{A}_{2}}^{U}-{F}_{{A}_{1}}^{U}{F}_{{A}_{2}}^{U}\right]>.$
- Subtraction${A}_{1}-{A}_{2}=<\left[{T}_{{A}_{1}}^{L}-{F}_{{A}_{2}}^{U},{T}_{{A}_{1}}^{U}-{F}_{{A}_{2}}^{L}\right],\left[\mathrm{max}\left({I}_{{A}_{1}}^{L},{I}_{{A}_{2}}^{l}\right),\mathrm{max}\left({I}_{{A}_{1}}^{U},{I}_{{A}_{2}}^{U}\right)\right],\left[{F}_{{A}_{1}}^{L}-{T}_{{A}_{2}}^{U},{F}_{{A}_{1}}^{U}-{T}_{{A}_{2}}^{L}\right].$
- Multiplication by a constant valueλ${A}_{1}=\left[1-{\left(1-{T}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},1-{\left(1-{T}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right],\left[{\left({I}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},{\left({I}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right],\left[{\left({F}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},{\left({F}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right]$,where λ >0.
- AdditionLet ${A}_{1},{A}_{2}$ two INSs where${A}_{1}=<\left[{T}_{{A}_{1}}^{L},{T}_{{A}_{1}}^{U}\right],\left[{I}_{{A}_{1}}^{L},{I}_{{A}_{1}}^{U}\right],\left[{F}_{{A}_{1}}^{L},{F}_{{A}_{1}}^{U}\right]>$,${A}_{2}=\left[{T}_{{A}_{2}}^{L},{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}\right],\left[{F}_{{A}_{2}}^{L},{F}_{{A}_{2}}^{U}\right]$ then ${A}_{1}+{A}_{2}=<\left[{T}_{{A}_{1}}^{L}+{T}_{{A}_{2}}^{L}-{T}_{{A}_{1}}^{L}{T}_{{A}_{2}}^{L},{T}_{{A}_{1}}^{U}+{T}_{{A}_{2}}^{U}-{T}_{{A}_{1}}^{U}{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{1}}^{L}{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}{I}_{{A}_{2}}^{U}\right],\left[{F}_{{A}_{1}}^{L}{F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}{F}_{{A}_{2}}^{U}\right]>.$
- Subset${A}_{1}\subseteq {A}_{2}$ if and only if ${T}_{{A}_{1}}^{L}\le {T}_{{A}_{2}}^{L}$,${T}_{{A}_{1}}^{U}\le {T}_{{A}_{2}}^{U}$; ${I}_{{A}_{1}}^{L}\ge {I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}\ge {I}_{{A}_{2}}^{U}$;${F}_{{A}_{1}}^{L}\ge {F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}\ge {F}_{{A}_{2}}^{U}$.
- Equality${A}_{1}={A}_{2}$ if and only if ${A}_{1}\subseteq {A}_{2}$ and ${A}_{2}\subseteq {A}_{1}$.
- ComplementLet $V=<\left[{T}_{V}^{L}\left(x\right),{T}_{V}^{U}\left(x\right)\right],\left[{I}_{V}^{L}\left(x\right),{I}_{V}^{U}\left(x\right)\right],\left[{F}_{V}^{L}\left(x\right),{F}_{V}^{U}\left(x\right)\right]>$,then ${V}^{c}$ = $<\left[{F}_{V}^{L}\left(x\right),{F}_{V}^{U}\left(x\right)\right],\left[1-{I}_{V}^{U}\left(x\right),1-{I}_{V}^{L}\left(x\right)\right],\left[{T}_{V}^{L}\left(x\right),{T}_{V}^{U}\left(x\right)\right]>$.
- Multiplication${A}_{1}\times {A}_{2}=<\left[{T}_{{A}_{1}}^{L}{T}_{{A}_{2}}^{L},{T}_{{A}_{1}}^{U}{T}_{{A}_{2}}^{U}\right],\left[{I}_{{A}_{1}}^{L}+{I}_{{A}_{2}}^{L}-{I}_{{A}_{1}}^{L}{I}_{{A}_{2}}^{L},{I}_{{A}_{1}}^{U}+{I}_{{A}_{2}}^{U}-{I}_{{A}_{1}}^{U}{I}_{{A}_{2}}^{U}\right],\left[{F}_{{A}_{1}}^{L}+{F}_{{A}_{2}}^{L}-{F}_{{A}_{1}}^{L}{F}_{{A}_{2}}^{L},{F}_{{A}_{1}}^{U}+{F}_{{A}_{2}}^{U}-{F}_{{A}_{1}}^{U}{F}_{{A}_{2}}^{U}\right]>.$
- Subtraction${A}_{1}-{A}_{2}=<\left[{T}_{{A}_{1}}^{L}-{F}_{{A}_{2}}^{U},{T}_{{A}_{1}}^{U}-{F}_{{A}_{2}}^{L}\right],\left[\mathrm{max}\left({I}_{{A}_{1}}^{L},{I}_{{A}_{2}}^{l}\right),\mathrm{max}\left({I}_{{A}_{1}}^{U},{I}_{{A}_{2}}^{U}\right)\right],\left[{F}_{{A}_{1}}^{L}-{T}_{{A}_{2}}^{U},{F}_{{A}_{1}}^{U}-{T}_{{A}_{2}}^{L}\right].$
- Multiplication by a constant valueλ${A}_{1}=\left[1-{\left(1-{T}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},1-{\left(1-{T}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right],\left[{\left({I}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},{\left({I}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right],\left[{\left({F}_{{A}_{1}}^{L}\right)}^{\mathsf{\lambda}},{\left({F}_{{A}_{1}}^{U}\right)}^{\mathsf{\lambda}}\right]$, where λ >0.

#### 3.3. Weighted Average for Interval-Valued Neutrosophic Numbers (INN)

#### 3.4. INS Deneutrosophication Function

#### 3.5. Ranking Method for Interval-Valued Neutrosophic Numbers

- if $D\left({A}_{1}\right)$ greater than $D\left({A}_{2}\right)$, then ${A}_{1}>{A}_{2};$
- if $D\left({A}_{1}\right)$ less than $D\left({A}_{2}\right)$, then ${A}_{1}<{A}_{2};$
- if $D\left({A}_{1}\right)$ equals $D\left({A}_{2}\right)$, then ${A}_{1}={A}_{2}$.

## 4. The ANP and TOPSIS Methods

#### 4.1. The Analytic Network Process (ANP)

- The decision-making problem should be structured as a network that consists of a main objective, criteria for achieving this objective and can be divided to sub-criteria, and finally all available alternatives. The feedback among network elements should be considered here.
- To calculate criteria’s and alternatives’ weights, the comparisons matrices should be constructed utilizing the 1–9 scale of Saaty. After then, we should check the consistency ratio of these matrices, and it must be $\le 0.1$ for each comparison matrix. The comparison matrix’s eigenvector should be calculated after that by summing up the columns of comparison matrix. A new matrix is constructed by dividing each value in a column by the summation of that column, and then taking the average of new matrix rows. For more information, see [51]. The ANP comparison matrices may be constructed for comparing:
- Criteria with respect to goal,
- Sub-criteria with respect to criterion from the same cluster,
- Alternatives with respect to each criterion,
- Criteria that belong to the same cluster with respect to each alternative.

- Use the eigenvectors calculated in the previous step for constructing the super-matrix columns. For obtaining a weighted super-matrix, a normalization process must be established. Then, raise the weighted matrix to a larger power until the raw values will be equal to each column values of super-matrix for obtaining the limiting matrix.
- Finally, choose the best alternative by depending on weight values.

#### 4.2. The TOPSIS Technique

- The decision makers should construct the evaluation matrix that consists of $m$ alternatives and $n$ criteria. The intersection of each alternative and criterion is denoted as ${x}_{ij}$, and then we have ${\left({x}_{ij}\right)}_{m*n}$ matrix.
- Use the following equation for obtaining the normalized evaluation matrix:$${\mathrm{r}}_{\mathrm{ij}}=\frac{{\mathrm{x}}_{\mathrm{ij}}}{\sqrt{{\sum}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{ij}}{}^{2}}};\text{}\mathrm{i}=1,\text{}2,\dots ,\mathrm{m};\text{}\mathrm{j}=1,\text{}2,\dots ,\mathrm{n}.$$
- Structure the weighted matrix through multiplying criteria’s weights ${\mathrm{w}}_{\mathrm{j}}$, by the normalized decision matrix ${\mathrm{r}}_{\mathrm{ij}}$ as follows:$${\mathrm{v}}_{\mathrm{ij}}={\mathrm{w}}_{\mathrm{j}}\times {\mathrm{r}}_{\mathrm{ij}}.$$
- Calculate the positive ${A}^{+}$ and negative ideal solution ${A}^{-}$ using the following:$${A}^{+}=\{<\mathrm{max}\left({v}_{ij}|i=1,2,\dots ,m\right)|j\in {J}^{+},\mathrm{min}\left({v}_{ij}|i=1,2,\dots ,m\right)|j\in {J}^{-}\},$$$${A}^{-}=\{<\mathrm{min}\left({v}_{ij}|i=1,2,\dots ,m\right)|j\in {J}^{+},\mathrm{max}\left({v}_{ij}|i=1,2,\dots ,m\right)|j\in {J}^{-}\},$$
- Calculate the Euclidean distance among positive (${d}_{i}^{+}$) and negative ideal solution (${d}_{i}^{-}$) as follows:$${d}_{i}^{+}=\sqrt{{\displaystyle \sum}_{j=1}^{n}{\left({v}_{ij}-{v}_{j}^{+}\right)}^{2}}i=1,2,\dots ,m,$$$${d}_{i}^{-}=\sqrt{{\displaystyle \sum}_{j=1}^{n}{\left({v}_{ij}-{v}_{j}^{-}\right)}^{2}}i=1,2,\dots ,m.$$
- Calculate the relative closeness to the ideal solution and make the final ranking of alternatives${c}_{i}=\frac{{d}_{i}^{-}}{{d}_{i}^{+}+{d}_{i}^{-}}$ for $i=1,2,\dots ,m$, and based on the largest ${\mathrm{c}}_{\mathrm{i}}$ value, begin to rank alternatives. (9)
- According to your rank of alternatives, take your final decision.

## 5. The Proposed Framework

**Phase 1:**For better understanding of a complex problem, we must firstly breakdown it.

- Step 1.1.
- Select a group of experts to share in making decisions. If we select $n$ experts, then we have the panel = [${e}_{1}$, ${e}_{2},$…, ${e}_{n}$].
- Step 1.2.
- Use the literature review to determine problem’s criteria and ask experts for confirming these criteria.
- Step 1.3.
- Determine the alternatives of the problem.
- Step 1.4.
- Begin to structure the hierarchy of the problem.

**Phase 2:**Calculate the weight of problem’s elements as follows:

- Step 2.1.
- The interval-valued comparison matrices should be constructed according to each expert and then aggregate experts’ matrices by using Equation (1).

- Step 2.2.
- Use the de-neutrosophication function for transforming the interval-valued neutrosophic numbers to crisp numbers as in Equation (2).
- Step 2.3.
- Use super decision software, which is available here (http://www.superdecisions.com/downloads/) to check the consistency of comparison matrices.
- Step 2.4.
- Calculate the eigenvectors for determining weight that will be used in building a super-matrix.
- Step 2.5.
- The super-matrix of interdependencies should be constructed after then.
- Step 2.6.
- Multiply the local weight, which was obtained from experts’ comparison matrices of criteria according to goal, by the weight of interdependence matrix of criteria for calculating global weight of criteria. In addition, calculate the global weights of sub-criteria by multiplying its local weight by the inner interdependent weight of the criterion to which it belongs.

**Phase 3:**Rank alternatives of problems.

- Step 3.1.
- Make the evaluation matrix, and then a normalization process must be performed for obtaining the normalized evaluation matrix using Equation (3).
- Step 3.2.
- Multiply criteria’s weights, which was obtained from ANP by the normalized evaluation matrix as in Equation (4) to construct the weighted matrix.
- Step 3.3.
- Determine positive and negative ideal solutions using Equations (5) and (6).
- Step 3.4.
- Calculate the Euclidean distance between positive solution (${d}_{i}^{+}$) and negative ideal solution (${d}_{i}^{-}$) using Equations (7) and (8).
- Step 3.5.
- Make the final ranking of alternatives based on closeness coefficient.

**Phase 4:**Compare the proposed method with other existing methods for validating it. The framework of the suggested method is presented in Figure 3.

## 6. The Case Study: Results and Analysis

**Phase 1:**Breakdown the complex problem for understanding it better.

**Phase 2:**Calculate the weights of problem elements.

**Phase 3:**Rank alternatives of problems.

- Step 3.6.
- Calculate the closeness coefficient using Equation (9), and make the final ranking of alternatives as in Table 21.

**Phase 4:**Validate the model and make comparisons with other existing methods.

## 7. Conclusions and Future Directions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Linguistic Variables | Interval-Valued Neutrosophic Numbers for Relative Importance <T,I,F> |
---|---|

Evenly significant | ([0.5,0.5], [0.5,0.5], [0.5,0.5]) |

Low significant | ([0.4,0.5], [0.1,0.2], [0.2,0.3]) |

Basically important | ([0.6,0.7], [0.0,0.1], [0.0,0.1]) |

Very strongly significant | ([0.8,0.9], [0.0,0.1], [0.0,0.1]) |

Absolutely significant | ([1,1], [0.0,0.1], [0.0,0.0]) |

Intermediate values | ([0.3,0.4], [0.1,0.2], [0.6,0.7]), |

([0.6,0.7], [0.1,0.2], [0.0,0.1]), | |

([0.7,0.8], [0.0,0.1], [0.0,0.1]), | |

([0.9,1], [0.0,0.1], [0.0,0.1]). |

Goal | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${\mathit{C}}_{1}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | [0.7,0.8], [0.0,0.1], [0.0,0.1] |

${\mathit{C}}_{2}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.6,0.7], [0.1,0.2], [0.0,0.1] | |

${\mathit{C}}_{3}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

Goal | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | Weights |
---|---|---|---|---|

${\mathit{C}}_{1}$ | 1 | 2 | 6 | 0.59 |

${\mathit{C}}_{2}$ | 0.5 | 1 | 4 | 0.32 |

${\mathit{C}}_{3}$ | 0.17 | 0.25 | 1 | 0.09 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|

${\mathit{C}}_{2}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.7,0.8], [0.0,0.1], [0.0,0.1] |

${\mathit{C}}_{3}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | Weights |
---|---|---|---|

${\mathit{C}}_{2}$ | 1 | 6 | 0.86 |

${\mathit{C}}_{3}$ | 0.17 | 1 | 0.14 |

${\mathit{C}}_{2}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{3}$ |
---|---|---|

${\mathit{C}}_{1}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.6,0.7], [0.1,0.2], [0.0,0.1] |

${\mathit{C}}_{3}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{2}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{3}$ | Weights |
---|---|---|---|

${\mathit{C}}_{1}$ | 1 | 4 | 0.8 |

${\mathit{C}}_{3}$ | 0.25 | 1 | 0.2 |

${\mathit{C}}_{3}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ |
---|---|---|

${\mathit{C}}_{1}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [1,1], [0.0,0.1], [0.0,0.0] |

${\mathit{C}}_{2}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{3}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | Weights |
---|---|---|---|

${\mathit{C}}_{1}$ | 1 | 9 | 0.9 |

${\mathit{C}}_{2}$ | 0.11 | 1 | 0.1 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | |
---|---|---|---|

${\mathit{C}}_{1}$ | 1 | 0.8 | 0.9 |

${\mathit{C}}_{2}$ | 0.86 | 1 | 0.1 |

${\mathit{C}}_{3}$ | 0.14 | 0.2 | 1 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | |
---|---|---|---|

${\mathit{C}}_{1}$ | 0.5 | 0.4 | 0.45 |

${\mathit{C}}_{2}$ | 0.43 | 0.5 | 0.05 |

${\mathit{C}}_{3}$ | 0.07 | 0.1 | 0.5 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ |
---|---|---|---|---|

${\mathit{C}}_{11}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.4,0.5], [0.1,0.2], [0.2,0.3] | [0.6,0.7], [0.1,0.2], [0.0,0.1] | [0.6,0.7], [0.0,0.1], [0.0,0.1] |

${\mathit{C}}_{12}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | [0.6,0.7], [0.1,0.2], [0.0,0.1] | |

${\mathit{C}}_{13}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | ||

${\mathit{C}}_{14}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | Weights |
---|---|---|---|---|---|

${\mathit{C}}_{11}$ | 1 | 3 | 4 | 5 | 0.54 |

${\mathit{C}}_{12}$ | 0.33 | 1 | 2 | 4 | 0.23 |

${\mathit{C}}_{13}$ | 0.25 | 0.50 | 1 | 2 | 0.13 |

${\mathit{C}}_{14}$ | 0.20 | 0.25 | 0.5 | 1 | 0.08 |

${\mathit{C}}_{2}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{24}$ |
---|---|---|---|---|

${\mathit{C}}_{21}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.4,0.5], [0.1,0.2], [0.2,0.3] | [0.8,0.9], [0.0,0.1], [0.0,0.1] | [1,1], [0.0,0.1], [0.0,0.0] |

${\mathit{C}}_{22}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.6,0.7], [0.0,0.1], [0.0,0.1] | [0.8,0.9], [0.0,0.1], [0.0,0.1] | |

${\mathit{C}}_{23}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | ||

${\mathit{C}}_{24}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{2}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{24}$ | Weights |
---|---|---|---|---|---|

${\mathit{C}}_{21}$ | 1 | 3 | 7 | 9 | 0.59 |

${\mathit{C}}_{22}$ | 0.33 | 1 | 5 | 7 | 0.29 |

${\mathit{C}}_{23}$ | 0.14 | 0.20 | 1 | 2 | 0.08 |

${\mathit{C}}_{24}$ | 0.11 | 0.14 | 0.50 | 1 | 0.05 |

${\mathit{C}}_{3}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ |
---|---|---|---|---|

${\mathit{C}}_{31}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | [0.4,0.5], [0.1,0.2], [0.2,0.3] | [1,1], [0.0,0.1], [0.0,0.0] |

${\mathit{C}}_{32}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.3,0.4], [0.1,0.2], [0.6,0.7] | [0.7,0.8], [0.0,0.1], [0.0,0.1] | |

${\mathit{C}}_{33}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] | [0.4,0.5], [0.1,0.2], [0.2,0.3] | ||

${\mathit{C}}_{34}$ | [0.5,0.5], [0.5,0.5], [0.5,0.5] |

${\mathit{C}}_{3}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ | Weights |
---|---|---|---|---|---|

${\mathit{C}}_{31}$ | 1 | 2 | 3 | 9 | 0.50 |

${\mathit{C}}_{32}$ | 0.50 | 1 | 2 | 6 | 0.29 |

${\mathit{C}}_{33}$ | 0.33 | 0.50 | 1 | 3 | 0.15 |

${\mathit{C}}_{34}$ | 0.11 | 0.17 | 0.33 | 1 | 0.05 |

Criteria Local Weight | Sub-Criteria | Local Weight | Global Weight |
---|---|---|---|

Economic factors (0.46) | ${C}_{11}$ | 0.54 | 0.25 |

${C}_{12}$ | 0.23 | 0.11 | |

${C}_{13}$ | 0.13 | 0.06 | |

${C}_{14}$ | 0.08 | 0.04 | |

Environmental factors (0.42) | ${C}_{21}$ | 0.59 | 0.25 |

${C}_{22}$ | 0.29 | 0.12 | |

${C}_{23}$ | 0.08 | 0.03 | |

${C}_{24}$ | 0.05 | 0.02 | |

Social factors (0.12) | ${C}_{31}$ | 0.50 | 0.06 |

${C}_{32}$ | 0.29 | 0.03 | |

${C}_{33}$ | 0.15 | 0.02 | |

${C}_{34}$ | 0.05 | 0.006 |

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{24}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{1}$ | 0.53 | 0.46 | 0.46 | 0.43 | 0.52 | 0.54 | 0.45 | 0.58 | 0.48 | 0.59 | 0.59 | 0.51 |

${\mathit{A}}_{2}$ | 0.46 | 0.58 | 0.53 | 0.48 | 0.43 | 0.58 | 0.59 | 0.52 | 0.54 | 0.54 | 0.46 | 0.64 |

${\mathit{A}}_{3}$ | 0.44 | 0.43 | 0.56 | 0.53 | 0.49 | 0.45 | 0.36 | 0.46 | 0.49 | 0.38 | 0.47 | 0.47 |

${\mathit{A}}_{4}$ | 0.56 | 0.52 | 0.43 | 0.55 | 0.54 | 0.41 | 0.56 | 0.43 | 0.48 | 0.45 | 0.46 | 0.32 |

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{24}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{1}$ | 0.13 | 0.05 | 0.03 | 0.02 | 0.13 | 0.06 | 0.01 | 0.01 | 0.03 | 0.02 | 0.01 | 0.003 |

${\mathit{A}}_{2}$ | 0.11 | 0.06 | 0.03 | 0.02 | 0.11 | 0.07 | 0.02 | 0.01 | 0.03 | 0.02 | 0.01 | 0.004 |

${\mathit{A}}_{3}$ | 0.11 | 0.05 | 0.03 | 0.02 | 0.12 | 0.05 | 0.01 | 0.01 | 0.03 | 0.01 | 0.01 | 0.003 |

${\mathit{A}}_{4}$ | 0.14 | 0.06 | 0.03 | 0.02 | 0.13 | 0.05 | 0.02 | 0.01 | 0.03 | 0.01 | 0.01 | 0.002 |

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

${\mathit{A}}_{1}$ | 0.020 | 0.032 | 0.615 | 2 |

${\mathit{A}}_{2}$ | 0.036 | 0.026 | 0.419 | 3 |

${\mathit{A}}_{3}$ | 0.041 | 0.010 | 0.196 | 4 |

${\mathit{A}}_{4}$ | 0.022 | 0.040 | 0.645 | 1 |

Alternatives | Weights | Rank |
---|---|---|

${\mathit{A}}_{1}$ | 0.245 | 3 |

${\mathit{A}}_{2}$ | 0.250 | 2 |

${\mathit{A}}_{3}$ | 0.244 | 4 |

${\mathit{A}}_{4}$ | 0.267 | 1 |

Alternatives | Weights | Rank |
---|---|---|

${\mathit{A}}_{1}$ | 0.26 | 1 |

${\mathit{A}}_{2}$ | 0.25 | 2 |

${\mathit{A}}_{3}$ | 0.23 | 3 |

${\mathit{A}}_{4}$ | 0.26 | 1 |

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{14}$ | ${\mathit{C}}_{21}$ | ${\mathit{C}}_{22}$ | ${\mathit{C}}_{23}$ | ${\mathit{C}}_{24}$ | ${\mathit{C}}_{31}$ | ${\mathit{C}}_{32}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{34}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{1}$ | 0.13 | 0.05 | 0.03 | 0.02 | 0.13 | 0.06 | 0.01 | 0.01 | 0.03 | 0.02 | 0.01 | 0.003 |

${\mathit{A}}_{2}$ | 0.11 | 0.06 | 0.03 | 0.02 | 0.11 | 0.07 | 0.02 | 0.01 | 0.03 | 0.02 | 0.01 | 0.004 |

${\mathit{A}}_{3}$ | 0.11 | 0.05 | 0.03 | 0.02 | 0.12 | 0.05 | 0.01 | 0.01 | 0.03 | 0.01 | 0.01 | 0.003 |

${\mathit{A}}_{4}$ | 0.14 | 0.06 | 0.03 | 0.02 | 0.13 | 0.05 | 0.02 | 0.01 | 0.03 | 0.01 | 0.01 | 0.002 |

${{\displaystyle \sum}}_{\mathit{j}=1}^{\mathit{g}}\mathit{x}\mathit{i}{\mathit{j}}^{*}$ | ${{\displaystyle \sum}}_{\mathit{j}=\mathit{g}+1}^{\mathit{n}}\mathit{x}\mathit{i}{\mathit{j}}^{*}$ | ${\mathit{p}}_{\mathit{i}}^{*}$ | Ranking | |
---|---|---|---|---|

${\mathit{A}}_{1}$ | 0.43 | 0.073 | 0.357 | 2 |

${\mathit{A}}_{2}$ | 0.41 | 0.084 | 0.326 | 4 |

${\mathit{A}}_{3}$ | 0.39 | 0.063 | 0.327 | 3 |

${\mathit{A}}_{4}$ | 0.44 | 0.072 | 0.368 | 1 |

${{\displaystyle \sum}}_{\mathit{j}=1}^{\mathit{g}}\mathit{x}\mathit{i}{\mathit{j}}^{*}$ | ${{\displaystyle \sum}}_{\mathit{j}=\mathit{g}+1}^{\mathit{n}}\mathit{x}\mathit{i}{\mathit{j}}^{*}$ | ${\mathit{p}}_{\mathit{i}}^{*}$ | Ranking | |
---|---|---|---|---|

${\mathit{A}}_{1}$ | 0.43 | 0.073 | 5.89 | 3 |

${\mathit{A}}_{2}$ | 0.41 | 0.084 | 4.88 | 4 |

${\mathit{A}}_{3}$ | 0.39 | 0.063 | 6.19 | 1 |

${\mathit{A}}_{4}$ | 0.44 | 0.072 | 6.11 | 2 |

Suppliers | Proposed Technique (1) | AHP (2) | ANP (3) | MOORA (4) | MOOSRA (5) |
---|---|---|---|---|---|

${\mathit{A}}_{1}$ | 2 | 3 | 1 | 2 | 3 |

${\mathit{A}}_{2}$ | 3 | 2 | 2 | 4 | 4 |

${\mathit{A}}_{3}$ | 4 | 4 | 3 | 3 | 1 |

${\mathit{A}}_{4}$ | 1 | 1 | 1 | 1 | 2 |

Correlation (1, 2) | Correlation (1, 3) | Correlation (1, 4) | Correlation (1, 5) |
---|---|---|---|

0.8 | 0.9 | 0.8 | 0.2 |

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## Share and Cite

**MDPI and ACS Style**

Abdel-Basset, M.; Mohamed, M.; Smarandache, F. A Hybrid Neutrosophic Group ANP-TOPSIS Framework for Supplier Selection Problems. *Symmetry* **2018**, *10*, 226.
https://doi.org/10.3390/sym10060226

**AMA Style**

Abdel-Basset M, Mohamed M, Smarandache F. A Hybrid Neutrosophic Group ANP-TOPSIS Framework for Supplier Selection Problems. *Symmetry*. 2018; 10(6):226.
https://doi.org/10.3390/sym10060226

**Chicago/Turabian Style**

Abdel-Basset, Mohamed, Mai Mohamed, and Florentin Smarandache. 2018. "A Hybrid Neutrosophic Group ANP-TOPSIS Framework for Supplier Selection Problems" *Symmetry* 10, no. 6: 226.
https://doi.org/10.3390/sym10060226