# Group Decision-Making Based on the VIKOR Method with Trapezoidal Bipolar Fuzzy Information

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Trapezoidal Bipolar Fuzzy VIKOR Method

#### 3.1. Identification of Linguistic Variables

#### 3.2. Construction of Decision Matrix

#### 3.3. Ranking of Bipolar Fuzzy Numbers

#### 3.4. Calculating the Normalized Weights

#### 3.5. Finding the Best and Worst Value

#### 3.6. Computing the Values ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$ and ${\mathcal{Q}}_{\alpha}$

#### 3.7. Ordering the Alternatives

#### 3.8. Compromising the Solution

## 4. Numerical Example

#### 4.1. Selection of Health-Care Waste Treatment Strategy

**Step****1.**- The group of decision-makers choose the linguistic variables to be assessed in the performance ratings of alternatives with respect to the criteria given in Figure 2 and Figure 3. The linguistic variables and their corresponding trapezoidal bipolar fuzzy numbers are presented for cost-type criteria and benefit-type criteria in Table 2 and Table 3, respectively.
**Step****2.**- These linguistic terms are used by decision-makers to determine the performance ratings of alternatives with respect to the conflicting criteria, which are shown in Table 4.

**Step****3.**- The aggregated bipolar fuzzy decision matrix is then converted into a simple decision matrix consisting of crisp values as entries by deploying the ranking function given in Equation (2) The decision matrix F is given in Table 7. For example, ${f}_{11}$ is calculated as,$$\begin{array}{cc}\hfill {f}_{11}& =\left(\right)open="("\; close=")">\left[\frac{0.6+0.77+0.8+1}{4}\right]+\left[\frac{-0.6-0.77+0.8+1}{2}\right]-\left(\right)open="("\; close=")">\left[\frac{0+0.17+0.27+0.3}{4}\right]+\left[\frac{-0-0.17+0.27+0.3}{2}\right]\hfill \end{array}\hfill \phantom{\rule{1.em}{0ex}}& =(0.793+0.215)-(0.185+0.2)=0.62.\hfill $$

**Step****4.**- The normalized weights for the criteria are calculated in this step by applying the entropy measure information. The projection values of the criteria are computed by using Equation (3), and the results are given in Table 8. For example, ${\mathbb{P}}_{11}$ is calculated as$$\begin{array}{c}\hfill {\mathbb{P}}_{11}=\frac{0.62}{0.62+0.37+0.29+0.09+0.43}=0.34.\end{array}$$These projection values are then used to enumerate the entropy value and the degree of divergence for criteria using Equations (4) and (5). Furthermore, the weight for each criterion is calculated by deploying the Equation (6), and results are respectively shown in Table 9. For instance, ${\mathbb{E}}_{1}$, ${\mathrm{d}}_{1}$, and ${\mathrm{W}}_{1}$ are calculated here, respectively, as$$\begin{array}{cc}{\mathbb{E}}_{1}\hfill & =-\frac{1}{log\left(5\right)}[0.34log\left(0.34\right)+0.21log\left(0.21\right)+0.16log\left(0.16\right)+0.05log\left(0.05\right)+0.24log\left(0.24\right)]\hfill \\ \hfill & =-\frac{1}{0.6989}(-0.6428)=0.92.\hfill \\ {\mathrm{d}}_{1}\hfill & =1-0.92=0.08.\hfill \\ {\mathrm{W}}_{1}\hfill & =\frac{0.08}{0.08+0.31+0.51+0.14+0.21}=\frac{0.08}{1.25}=0.064.\hfill \end{array}$$

**Step****5.**- The best or ideal value ${f}_{\beta}^{*}$, as well as the worst or nadir value ${f}_{\beta}^{-}$, for cost-type and benefit-type criteria are determined by using Equations (7) and (8), respectively, and the results are shown in Table 10.

**Step****6.**- The values of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$ are given in Table 11. Here, the value of the weight of the strategy “$\upsilon $” is 0.5.

**Step****7.**- The ranking of waste treatment strategies according to the ascending orders of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$ is shown in Table 12.

**Step****8.**- The two conditions mentioned in Section 3.8 are satisfied, and thus the alternative ${\mathcal{T}}_{2}$, that is, Pyrolytic incineration, is the best possible waste treatment strategy.

#### 4.2. Selection of Site for Thermal Power Station

**Step****1.**- The group of decision-makers can then decide to use the linguistic variables for evaluating the performance values of alternatives with respect to the criteria. The linguistic variables and their corresponding trapezoidal bipolar fuzzy numbers are given in Table 13 and Table 14 for cost- and benefit-type criteria, respectively.

**Step****2.**- The preference ratings of alternatives on the basis of conflicting criteria given by decision-makers in the form of linguistic variables are given in Table 15. These linguistic variables are then converted into corresponding trapezoidal bipolar fuzzy numbers by using Table 13 and Table 14, and the results are shown in Table 16. By using these trapezoidal bipolar fuzzy numbers, an aggregated decision matrix is constructed by employing the expressions given in Equation (1), and is shown in Table 17.

**Step****3.****Step****4.**- The normalized weights for the criteria are calculated in this step by applying the entropy measure information. The projection value of each criterion is computed by using Equation (3), and the results are given in Table 19. These projection values are then used to enumerate the entropy value and the degree of divergence for the criteria using Equations (4) and (5). Furthermore, the weight for each criterion is calculated by deploying the Equation (6), and the results are respectively shown in Table 20.

**Step****5.**- The best or ideal value ${f}_{\beta}^{*}$, as well as the worst value ${f}_{\beta}^{-}$ for cost-type and benefit-type criteria is determined by using Equations (7) and (8), respectively, and the results are shown in Table 21.

**Step****6.**- The values of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$ are given in Table 22. Here, the value of the weight of strategy “$\upsilon $” is taken as 0.5.

**Step****7.**- The ranking of waste treatment strategies according to the ascending orders of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$ is shown in Table 23.

**Step****8.**- The two conditions mentioned in Section 3.8 are satisfied, and thus the location ${\mathcal{T}}_{2}$ is the best possible site for planting a thermal power station.

## 5. Comparative Analysis with Trapezoidal Bipolar Fuzzy TOPSIS

#### Trapezoidal Bipolar Fuzzy TOPSIS

## 6. Comparison of Trapezoidal Bipolar Fuzzy VIKOR with Fuzzy VIKOR

- Bipolar fuzzy sets improve the performance of well-established structures for decision-making, because they provide two-sided information for evaluating the alternatives with respect to each criteria. Bipolarity is essential to recognize positive data, which specifies what is ensured to be conceivable, as well as negative data, which specifies what is prohibited, or most likely false. In a different interpretation, two-sided information appears in the context of necessary and possible consequences [28]. Bipolar fuzzy numbers, as an extension of bipolar fuzzy subsets, are defined on the real line, and they can be used more appropriately for decision-making. The linguistic terms or values are induced by trapezoidal bipolar fuzzy linguistic variables, in which the interval for a positive membership degree shows the satisfaction behavior of an alternative towards a criterion. On the other hand, the interval for a negative membership degree represents the dissatisfaction of that alternative based on the criteria. We have used the trapezoidal bipolar fuzzy VIKOR approaches for evaluation, or to obtain the compromise solution. Our motivation lies in real-world problems where we find two-sided (instead of one-sided) information, as we have described in the examples.
- Crisp or fuzzy sets only provide us with one-sided information for making decisions. Put differently, we can say that we only have the information about the satisfaction degree of the alternatives under the corresponding criteria. By using these set structures, we are unable to take advantage of any information about the dissatisfaction degree of these alternatives with respect to the conflicting criteria. We have thus presented the trapezoidal bipolar fuzzy VIKOR method for a compromising solution as an improvement of previous successful versions of the VIKOR method.

## 7. Insights of This Method

- Our version of the VIKOR method is a generalized form of other existing VIKOR methods, as it first deals with bipolar fuzzy information or data.
- The Shannon entropy concept for weights calculation is used to avoid the personal interest or bias of the decision-makers towards the criteria.
- Linguistic variables parameterized by trapezoidal bipolar fuzzy numbers are used instead of bipolar fuzzy sets, since they provide more suitable results.
- Two different applications using this method are discussed.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Authors | Year | Contributions |
---|---|---|

Zhang [19] | 1994 | Initiated the concept of bipolar fuzzy sets. |

Opricovic, Tzeng [3] | 1998 | Introduce the VIKOR method. |

Opricovic, Tzeng [12] | 2007 | Extended VIKOR method in comparison with outranking method. |

Sanayei et al. [13] | 2010 | Introduce the fuzzy VIKOR method for group decision-making. |

Shemshadi et al. [14] | 2011 | Application of fuzzy VIKOR using entropy weight information. |

Luo, Wang [18] | 2017 | Present an intuitionistic fuzzy VIKOR method. |

Akram, Arshad [22] | 2019 | Introduce the bipolar fuzzy numbers. |

Shumaiza et al. | This paper | Present the trapezoidal bipolar fuzzy VIKOR method. |

Linguistic Variable | Abbreviation | Bipolar Fuzzy Number |
---|---|---|

$Low$ | L | $\langle (0.0,0.0,0.1,0.2),(0.7,0.8,0.9,1.0)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}low$ | $ML$ | $\langle (0.1,0.2,0.3,0.4),(0.6,0.7,0.8,0.8)\rangle $ |

$Medium$ | M | $\langle (0.4,0.5,0.5,0.6),(0.5,0.5,0.6,0.7)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}high$ | $MH$ | $\langle (0.6,0.7,0.7,0.8),(0.1,0.2,0.3,0.3)\rangle $ |

$High$ | H | $\langle (0.8,0.9,1.0,1.0),(0.0,0.1,0.2,0.2)\rangle $ |

Linguistic Variable | Abbreviation | Bipolar Fuzzy Number |
---|---|---|

$Poor$ | P | $\langle (0.0,0.0,0.1,0.2),(0.7,0.8,0.9,1.0)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}Poor$ | $MP$ | $\langle (0.1,0.2,0.3,0.4),(0.6,0.7,0.8,0.8)\rangle $ |

$Fair$ | F | $\langle (0.4,0.5,0.5,0.6),(0.5,0.5,0.6,0.7)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}good$ | $MG$ | $\langle (0.6,0.7,0.7,0.8),(0.1,0.2,0.3,0.3)\rangle $ |

$Good$ | G | $\langle (0.8,0.9,1.0,1.0),(0.0,0.1,0.2,0.2)\rangle $ |

${\mathbb{D}}_{1}$ | ${\mathbb{D}}_{2}$ | ${\mathbb{D}}_{3}$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | ${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | ${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | |

${\mathcal{K}}_{1}$ | $MG$ | G | G | $MG$ | $MG$ | G | F | $MG$ | F | $MG$ | $MG$ | G | F | $MG$ | $MG$ |

${\mathcal{K}}_{2}$ | F | G | F | $MG$ | $MG$ | F | G | $MG$ | G | F | $MG$ | G | G | $MG$ | F |

${\mathcal{K}}_{3}$ | M | $MG$ | H | $MH$ | M | $MH$ | M | M | H | $MG$ | M | M | H | H | M |

${\mathcal{K}}_{4}$ | G | G | F | $MG$ | $MG$ | $MG$ | G | F | G | $MG$ | G | $MG$ | G | G | $MG$ |

${\mathcal{K}}_{5}$ | $MH$ | H | $MH$ | H | M | $ML$ | M | H | $MH$ | $MH$ | $MH$ | M | $MH$ | $MH$ | $MH$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | ||
---|---|---|---|---|---|---|

${\mathbb{D}}_{1}$ | ${\mathcal{T}}_{1}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | |

$(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | ||

${\mathbb{D}}_{2}$ | ${\mathcal{T}}_{1}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.1,0.2,0.3,0.4),$ |

$(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.6,0.7,0.8,0.8)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle \left(\right(0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ | |

$(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathbb{D}}_{3}$ | ${\mathcal{T}}_{1}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.1,0.2,0.3,0.4),$ | |

$(0.5,0.5,0.6,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.6,0.7,0.8,0.8)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.6,0.7,0.7,0.8),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.5,0.5,0.6,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $\langle (0.6,0.77,0.80,1.0),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.6,0.83,0.9,1.0),$ | $\langle (0.1,0.53,0.57,0.8),$ |

$(0.0,0.17,0.27,0.3)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.0,0.13,0.23,0.3)\rangle $ | $(0.1,0.37,0.47,0.8)\rangle $ | |

${\mathcal{T}}_{2}$ | $\langle (0.4,0.77,0.83,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.6,0.83,0.9,1.0),$ | $\langle (0.4,0.63,0.67,1.0),$ |

$(0.0,0.23,0.33,0.7)\rangle $ | $(0.0,0.1,0.2,0.2)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.0,0.13,0.23,0.3)\rangle $ | $(0.0,0.37,0.47,0.7)\rangle $ | |

${\mathcal{T}}_{3}$ | $\langle (0.4,0.7,0.73,1.0),$ | $\langle (0.4,0.7,0.73,1.0),$ | $\langle (0.4,0.77,0.83,1.0),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.1,0.6,0.67,1.0),$ |

$(0.0,0.27,0.37,0.7)\rangle $ | $(0.0,0.27,0.37,0.7)\rangle $ | $(0.0,0.23,0.33,0.7)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.0,0.33,0.43,0.8)\rangle $ | |

${\mathcal{T}}_{4}$ | $\langle (0.4,0.63,0.63,0.8),$ | $\langle (0.6,0.77,0.8,1.0),$ | $\langle (0.6,0.83,0.9,1.0),$ | $\langle (0.6,0.83,0.9,1.0),$ | $\langle (0.6,0.77,0.8,1.0),$ |

$(0.1,0.3,0.4,0.7)\rangle $ | $(0.0,0.17,0.27,0.3)\rangle $ | $(0.0,0.13,0.23,0.3)\rangle $ | $(0.0,0.13,0.23,0.3)\rangle $ | $(0.0,0.17,0.27,0.3)\rangle $ | |

${\mathcal{T}}_{5}$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.4,0.57,0.57,0.8),$ | $\langle (0.6,0.7,0.7,0.8),$ | $\langle (0.4,0.63,0.63,0.8),$ |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.1,0.4,0.5,0.7)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.3,0.4,0.7)\rangle $ |

F | ${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $0.62$ | $0.01$ | $0.01$ | $0.70$ | $0.04$ |

${\mathcal{T}}_{2}$ | $0.37$ | $0.80$ | $0.01$ | $0.70$ | $0.21$ |

${\mathcal{T}}_{3}$ | $0.29$ | $0.29$ | $0.37$ | $0.01$ | $0.24$ |

${\mathcal{T}}_{4}$ | $0.09$ | $0.60$ | $0.70$ | $0.70$ | $0.62$ |

${\mathcal{T}}_{5}$ | $0.43$ | $0.01$ | $0.01$ | $0.43$ | $0.09$ |

${\mathbb{P}}_{\mathit{\alpha}\mathit{\beta}}$ | ${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $0.34$ | $0.01$ | $0.01$ | $0.28$ | $0.03$ |

${\mathcal{T}}_{2}$ | $0.21$ | $0.46$ | $0.01$ | $0.28$ | $0.18$ |

${\mathcal{T}}_{3}$ | $0.16$ | $0.17$ | $0.34$ | $0.003$ | $0.2$ |

${\mathcal{T}}_{4}$ | $0.05$ | $0.35$ | $0.64$ | $0.28$ | $0.52$ |

${\mathcal{T}}_{5}$ | $0.24$ | $0.01$ | $0.01$ | $0.17$ | $0.08$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${\mathbb{E}}_{\beta}$ | $0.92$ | $0.69$ | $0.49$ | $0.86$ | $0.79$ |

${\mathrm{d}}_{\beta}$ | $0.08$ | $0.31$ | $0.51$ | $0.14$ | $0.21$ |

${\mathrm{W}}_{\beta}$ | $0.064$ | $0.248$ | $0.408$ | $0.112$ | $0.168$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${f}_{\beta}^{*}$ | $0.62$ | $0.80$ | $0.01$ | $0.70$ | $0.04$ |

${f}_{\beta}^{-}$ | $0.09$ | $0.01$ | $0.70$ | $0.01$ | $0.62$ |

**Table 11.**Values of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$.

${\mathcal{S}}_{\mathit{\alpha}}$ | ${\mathcal{R}}_{\mathit{\alpha}}$ | ${\mathcal{Q}}_{\mathit{\alpha}}$ | |
---|---|---|---|

${\mathcal{T}}_{1}$ | $0.248$ | $0.249$ | $0.423$ |

${\mathcal{T}}_{2}$ | $0.079$ | $0.049$ | $0.0$ |

${\mathcal{T}}_{3}$ | $0.583$ | $0.213$ | $0.646$ |

${\mathcal{T}}_{4}$ | $0.703$ | $0.408$ | $1.0$ |

${\mathcal{T}}_{5}$ | $0.329$ | $0.248$ | $0.488$ |

**Table 12.**The ranking of alternatives by ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

By ${\mathcal{S}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{4}$ |

By ${\mathcal{R}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{4}$ |

By ${\mathcal{Q}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{4}$ |

Linguistic Variable | Abbreviation | Bipolar Fuzzy Number |
---|---|---|

$Very\phantom{\rule{3.33333pt}{0ex}}low$ | $VL$ | $\langle (0.0,0.0,0.1,0.2),(0.8,0.9,1.0,1.0)\rangle $ |

$Low$ | L | $\langle (0.1,0.2,0.2,0.3),(0.7,0.8,0.8,0.9)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}low$ | $ML$ | $\langle (0.2,0.3,0.4,0.5),(0.5,0.6,0.7,0.8)\rangle $ |

$Medium$ | M | $\langle (0.4,0.5,0.5,0.6),(0.4,0.5,0.5,0.6)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}high$ | $MH$ | $\langle (0.5,0.6,0.7,0.8),(0.2,0.3,0.4,0.5)\rangle $ |

$High$ | H | $\langle (0.7,0.8,0.8,0.9),(0.1,0.2,0.3,0.3)\rangle $ |

$Very\phantom{\rule{3.33333pt}{0ex}}high$ | $VH$ | $\langle (0.8,0.9,1.0,1.0),(0.0,0.0,0.1,0.2)\rangle $ |

Linguistic Variable | Abbreviation | Bipolar Fuzzy Number |
---|---|---|

$Very\phantom{\rule{3.33333pt}{0ex}}poor$ | $VP$ | $\langle (0.0,0.0,0.1,0.2),(0.8,0.9,1.0,1.0)\rangle $ |

$Poor$ | P | $\langle (0.1,0.2,0.2,0.3),(0.7,0.8,0.8,0.9)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}Poor$ | $MP$ | $\langle (0.2,0.3,0.4,0.5),(0.5,0.6,0.7,0.8)\rangle $ |

$Fair$ | F | $\langle (0.4,0.5,0.5,0.6),(0.4,0.5,0.5,0.6)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}good$ | $MG$ | $\langle (0.5,0.6,0.7,0.8),(0.2,0.3,0.4,0.5)\rangle $ |

$Good$ | G | $\langle (0.7,0.8,0.8,0.9),(0.1,0.2,0.3,0.3)\rangle $ |

$Very\phantom{\rule{3.33333pt}{0ex}}Good$ | $VG$ | $\langle (0.8,0.9,1.0,1.0),(0.0,0.0,0.1,0.2)\rangle $ |

${\mathbb{D}}_{1}$ | ${\mathbb{D}}_{2}$ | ${\mathbb{D}}_{3}$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | ${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | ${\mathcal{T}}_{\mathbf{1}}$ | ${\mathcal{T}}_{\mathbf{2}}$ | ${\mathcal{T}}_{\mathbf{3}}$ | ${\mathcal{T}}_{\mathbf{4}}$ | ${\mathcal{T}}_{\mathbf{5}}$ | |

${\mathcal{K}}_{1}$ | $MG$ | G | G | $MG$ | $MG$ | G | $VG$ | $MG$ | F | $MG$ | $MG$ | $VG$ | F | $MG$ | $MG$ |

${\mathcal{K}}_{2}$ | F | $VG$ | F | $MG$ | $MG$ | F | G | $MG$ | G | F | $MG$ | G | G | $MG$ | F |

${\mathcal{K}}_{3}$ | M | $ML$ | $MH$ | $MH$ | M | $MH$ | $MG$ | M | M | $MG$ | H | M | H | M | M |

${\mathcal{K}}_{4}$ | G | $VG$ | F | $MG$ | $MG$ | $MG$ | $VG$ | F | G | $MG$ | G | G | $MG$ | $MG$ | $MG$ |

${\mathcal{K}}_{5}$ | $MH$ | $MH$ | H | $VH$ | H | $ML$ | M | H | $VH$ | $MH$ | $MH$ | M | $ML$ | $MH$ | $MH$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | ||
---|---|---|---|---|---|---|

${\mathbb{D}}_{1}$ | ${\mathcal{T}}_{1}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.2,0.3,0.4,0.5),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.1,0.2,0.2,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.5,0.6,0.7,0.8)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.1,0.2,0.2,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | ||

${\mathbb{D}}_{2}$ | ${\mathcal{T}}_{1}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.2,0.3,0.4,0.5),$ |

$(0.1,0.2,0.2,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.5,0.6,0.7,0.8)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathbb{D}}_{3}$ | ${\mathcal{T}}_{1}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{T}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | ||

${\mathcal{T}}_{3}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.2,0.3,0.4,0.5),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.1,0.2,0.2,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.5,0.6,0.7,0.8)\rangle $ | ||

${\mathcal{T}}_{4}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{T}}_{5}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $\langle (0.5,0.67,0.73,0.9),$ | $\langle (0.4,0.53,0.57,0.8),$ | $\langle (0.4,0.63,0.67,0.9),$ | $\langle (0.5,0.73,0.76,0.9),$ | $\langle (0.2,0.5,0.6,0.8),$ |

$(0.1,0.27,0.33,0.5)\rangle $ | $(0.2,0.43,0.47,0.6)\rangle $ | $(0.1,0.33,0.37,0.6)\rangle $ | $(0.1,0.23,0.27,0.5)\rangle $ | $(0.2,0.4,0.5,0.8)\rangle $ | |

${\mathcal{T}}_{2}$ | $\langle (0.7,0.87,0.93,1.0),$ | $\langle (0.7,0.83,0.87,1.0),$ | $\langle (0.2,0.43,0.47,0.6),$ | $\langle (0.7,0.87,0.93,1.0),$ | $\langle (0.4,0.53,0.57,0.8),$ |

$(0.1,0.07,0.13,0.3)\rangle $ | $(0.0,0.13,0.17,0.3)\rangle $ | $(0.4,0.53,0.57,0.8)\rangle $ | $(0.0,0.07,0.13,0.3)\rangle $ | $(0.2,0.43,0.47,0.6)\rangle $ | |

${\mathcal{T}}_{3}$ | $\langle (0.4,0.63,0.67,0.9),$ | $\langle (0.4,0.63,0.67,0.9),$ | $\langle (0.4,0.63,0.67,0.9),$ | $\langle (0.4,0.53,0.57,0.8),$ | $\langle (0.2,0.63,0.67,0.9),$ |

$(0.1,0.33,0.37,0.6)\rangle $ | $(0.1,0.33,0.37,0.6)\rangle $ | $(0.1,0.33,0.37,0.6)\rangle $ | $(0.2,0.43,0.47,0.6)\rangle $ | $(0.1,0.33,0.37,0.8)\rangle $ | |

${\mathcal{T}}_{4}$ | $\langle (0.4,0.57,0.63,0.8),$ | $\langle (0.5,0.67,0.73,0.9),$ | $\langle (0.4,0.53,0.57,0.8),$ | $\langle (0.5,0.67,0.73,0.9),$ | $\langle (0.5,0.8,0.9,1.0),$ |

$(0.2,0.37,0.43,0.6)\rangle $ | $(0.1,0.27,0.33,0.5)\rangle $ | $(0.2,0.43,0.47,0.6)\rangle $ | $(0.1,0.27,0.33,0.5)\rangle $ | $(0.0,0.1,0.2,0.5)\rangle $ | |

${\mathcal{T}}_{5}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.2,0.43,0.47,0.6),$ | $\langle (0.2,0.43,0.47,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.67,0.73,0.9),$ |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.53,0.57,0.8)\rangle $ | $(0.4,0.53,0.57,0.8)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.27,0.33,0.5)\rangle $ |

F | ${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $0.40$ | $0.15$ | $0.30$ | $0.44$ | $0.05$ |

${\mathcal{T}}_{2}$ | $0.75$ | $0.70$ | $0.15$ | $0.75$ | $0.15$ |

${\mathcal{T}}_{3}$ | $0.30$ | $0.30$ | $0.30$ | $0.15$ | $0.20$ |

${\mathcal{T}}_{4}$ | $0.20$ | $0.40$ | $0.15$ | $0.40$ | $0.60$ |

${\mathcal{T}}_{5}$ | $0.30$ | $0.15$ | $0.15$ | $0.30$ | $0.40$ |

${\mathbb{P}}_{\mathit{\alpha}\mathit{\beta}}$ | ${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ |
---|---|---|---|---|---|

${\mathcal{T}}_{1}$ | $0.21$ | $0.09$ | $0.29$ | $0.22$ | $0.04$ |

${\mathcal{T}}_{2}$ | $0.38$ | $0.41$ | $0.14$ | $0.37$ | $0.11$ |

${\mathcal{T}}_{3}$ | $0.15$ | $0.18$ | $0.29$ | $0.07$ | $0.14$ |

${\mathcal{T}}_{4}$ | $0.10$ | $0.24$ | $0.14$ | $0.20$ | $0.43$ |

${\mathcal{T}}_{5}$ | $0.15$ | $0.09$ | $0.14$ | $0.15$ | $0.29$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${\mathbb{E}}_{\beta}$ | $0.93$ | $0.90$ | $0.96$ | $0.93$ | $0.85$ |

${\mathrm{d}}_{\beta}$ | $0.07$ | $0.10$ | $0.04$ | $0.07$ | $0.15$ |

${\mathrm{W}}_{\beta}$ | $0.16$ | $0.23$ | $0.09$ | $0.16$ | $0.35$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | ${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|---|---|---|

${f}_{\beta}^{*}$ | $0.75$ | $0.70$ | $0.15$ | $0.75$ | $0.05$ |

${f}_{\beta}^{-}$ | $0.02$ | $0.15$ | $0.30$ | $0.15$ | $0.60$ |

**Table 22.**Values of ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$.

${\mathcal{S}}_{\mathit{\alpha}}$ | ${\mathcal{R}}_{\mathit{\alpha}}$ | ${\mathcal{Q}}_{\mathit{\alpha}}$ | |
---|---|---|---|

${\mathcal{T}}_{1}$ | $0.5045$ | $0.230$ | $0.6219$ |

${\mathcal{T}}_{2}$ | $0.0636$ | $0.0636$ | $0.0$ |

${\mathcal{T}}_{3}$ | $0.6437$ | $0.1673$ | $0.617$ |

${\mathcal{T}}_{4}$ | $0.7288$ | $0.350$ | $1.0$ |

${\mathcal{T}}_{5}$ | $0.7036$ | $0.235$ | $0.7716$ |

**Table 23.**The ranking of alternatives by ${\mathcal{S}}_{\alpha}$, ${\mathcal{R}}_{\alpha}$, and ${\mathcal{Q}}_{\alpha}$.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

By ${\mathcal{S}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{4}$ |

By ${\mathcal{R}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{4}$ |

By ${\mathcal{Q}}_{\alpha}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{T}}_{1}$ | ${\mathcal{T}}_{5}$ | ${\mathcal{T}}_{4}$ |

${\mathcal{K}}_{1}$ | ${\mathcal{K}}_{2}$ | ${\mathcal{K}}_{3}$ | |
---|---|---|---|

${\mathcal{T}}_{1}$ | $\langle (0.038,0.049,0.051,0.064),$ | $\langle (0.099,0.141,0.141,0.198),$ | $\langle (0.163,0.233,0.233,0.326),$ |

$(0.0,0.011,0.017,0.019)\rangle $ | $(0.025,0.099,0.124,0.174)\rangle $ | $(0.041,0.163,0.204,0.286)\rangle $ | |

${\mathcal{T}}_{2}$ | $\langle (0.026,0.049,0.053,0.064),$ | $\langle (0.198,0.223,0.248,0.248),$ | $\langle (0.163,0.233,0.233,0.326),$ |

$(0.0,0.015,0.021,0.045)\rangle $ | $(0.0,0.025,0.05,0.05)\rangle $ | $(0.041,0.163,0.204,0.286)\rangle $ | |

${\mathcal{T}}_{3}$ | $\langle (0.026,0.045,0.047,0.064),$ | $\langle (0.099,0.174,0.181,0.248),$ | $\langle (0.163,0.314,0.339,0.408),$ |

$(0.0,0.017,0.024,0.045)\rangle $ | $(0.0,0.067,0.092,0.174)\rangle $ | $(0.0,0.094,0.135,0.286)\rangle $ | |

${\mathcal{T}}_{4}$ | $\langle (0.026,0.04,0.04,0.051),$ | $\langle (0.149,0.191,0.198,0.248),$ | $\langle (0.245,0.339,0.367,0.408),$ |

$(0.006,0.019,0.026,0.045)\rangle $ | $(0.0,0.042,0.067,0.074)\rangle $ | $(0.0,0.053,0.094,0.122)\rangle $ | |

${\mathcal{T}}_{5}$ | $\langle (0.038,0.045,0.045,0.051),$ | $\langle (0.099,0.141,0.141,0.198),$ | $\langle (0.163,0.233,0.233,0.326),$ |

$(0.006,0.013,0.019,0.019)\rangle $ | $(0.025,0.099,0.124,0.174)\rangle $ | $(0.041,0.163,0.204,0.286)\rangle $ |

${\mathcal{K}}_{4}$ | ${\mathcal{K}}_{5}$ | |
---|---|---|

${\mathcal{T}}_{1}$ | $\langle (0.067,0.093,0.101,0.112),$ | $\langle (0.017,0.089,0.096,0.134),$ |

$(0.0,0.015,0.026,0.034)\rangle $ | $(0.017,0.062,0.079,0.134)\rangle $ | |

${\mathcal{T}}_{2}$ | $\langle (0.067,0.093,0.101,0.112),$ | $\langle (0.067,0.106,0.113,0.168),$ |

$(0.0,0.015,0.026,0.034)\rangle $ | $(0.0,0.062,0.079,0.118)\rangle $ | |

${\mathcal{T}}_{3}$ | $\langle (0.045,0.064,0.064,0.09),$ | $\langle (0.017,0.101,0.113,0.168),$ |

$(0.011,0.049,0.056,0.078)\rangle $ | $(0.0,0.055,0.072,0.134)\rangle $ | |

${\mathcal{T}}_{4}$ | $\langle (0.067,0.093,0.101,0.112),$ | $\langle (0.101,0.129,0.134,0.168),$ |

$(0.0,0.015,0.026,0.034)\rangle $ | $(0.0,0.029,0.045,0.05)\rangle $ | |

${\mathcal{T}}_{5}$ | $\langle (0.067,0.078,0.078,0.09),$ | $\langle (0.067,0.106,0.106,0.134),$ |

$(0.011,0.022,0.034,0.034)\rangle $ | $(0.017,0.05,0.067,0.118)\rangle $ |

Alternatives | Distance from BFPIS (${\mathit{E}}_{\mathit{\alpha}}^{+}$) | Distance from BFNIS (${\mathit{E}}_{\mathit{\beta}}^{-}$) | Closeness Coefficient |
---|---|---|---|

${\mathcal{T}}_{1}$ | $0.169$ | $0.250$ | $0.597$ |

${\mathcal{T}}_{2}$ | $0.053$ | $0.291$ | $0.846$ |

${\mathcal{T}}_{3}$ | $0.202$ | $0.178$ | $0.468$ |

${\mathcal{T}}_{4}$ | $0.258$ | $0.133$ | $0.340$ |

${\mathcal{T}}_{5}$ | $0.177$ | $0.234$ | $0.569$ |

Methods | Final Results of Alternatives | Ranking of Alternatives |
---|---|---|

Trapezoidal bipolar fuzzy TOPSIS | $0.597$, $0.846$, $0.468$, $0.340$, $0.569$ | ${\mathcal{T}}_{2}\succ {\mathcal{T}}_{1}\succ {\mathcal{T}}_{5}\succ {\mathcal{T}}_{3}\succ {\mathcal{T}}_{4}$ |

Trapezoidal bipolar fuzzy VIKOR | $0.423$, $0.0$, $0.646$, $1.0$, $0.488$ | ${\mathcal{T}}_{2}\succ {\mathcal{T}}_{1}\succ {\mathcal{T}}_{5}\succ {\mathcal{T}}_{3}\succ {\mathcal{T}}_{4}$ |

© 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**

Shumaiza; Akram, M.; Al-Kenani, A.N.; Alcantud, J.C.R.
Group Decision-Making Based on the VIKOR Method with Trapezoidal Bipolar Fuzzy Information. *Symmetry* **2019**, *11*, 1313.
https://doi.org/10.3390/sym11101313

**AMA Style**

Shumaiza, Akram M, Al-Kenani AN, Alcantud JCR.
Group Decision-Making Based on the VIKOR Method with Trapezoidal Bipolar Fuzzy Information. *Symmetry*. 2019; 11(10):1313.
https://doi.org/10.3390/sym11101313

**Chicago/Turabian Style**

Shumaiza, Muhammad Akram, Ahmad N. Al-Kenani, and José Carlos R. Alcantud.
2019. "Group Decision-Making Based on the VIKOR Method with Trapezoidal Bipolar Fuzzy Information" *Symmetry* 11, no. 10: 1313.
https://doi.org/10.3390/sym11101313