# Novel Extension of DEMATEL Method by Trapezoidal Fuzzy Numbers and D Numbers for Management of Decision-Making Processes

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. D Numbers

**Definition**

**1**

**.**Let $\Upsilon $ be a finite nonempty set, and a D number is a mapping that $D:\Upsilon \to [0,1]$, with:

**Definition**

**2**

**.**Let two D numbers ${D}_{1}=\left\{({b}_{1},{v}_{1}),\dots ,({b}_{i},{v}_{i}),\dots ,({b}_{n},{v}_{n})\right\}$ and ${D}_{2}=\left\{({b}_{n},{v}_{n}),\dots ,({b}_{i},{v}_{i}),\dots ,({b}_{1},{v}_{1})\right\}$ $({b}_{i},{v}_{i})$, $({b}_{j},{v}_{j})$ … $({b}_{n},{v}_{n})\}$ be given. Then, we can define the rule for the combination of D numbers $D={D}_{1}\odot {D}_{2}$ as follows:

_{1}and D

_{2}are defined in the frame of discernment and if ${Q}_{1}=1$ and ${Q}_{2}=1$, then the rule of combining D numbers (Rule (3)) is transformed into Dempster’s rule. Rule (3) of numbers is an algorithm for the combination and fusion of uncertain information presented in D numbers.

#### 2.2. Fuzzy Set Theory

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- (1)
- Addition:

- (2)
- Multiplication:

- (3)
- Subtraction:

- (4)
- Division:

- (5)
- Reciprocal values:

## 3. TrFN DEMATEL-D Methodology

_{i}represents the sum of the ith raw of the matrix T. The determined value presents the total direct and indirect effects that the criterion i provides for the other criteria. Meanwhile, the value of C

_{i}represents the sum of the jth column of the matrix T and shows the effects that the criterion j receives from the other criteria [37].

## 4. Application of TrFN D-DEMATEL Method

_{1}represents the experts’ preferences of the EG1 expert group and D

_{2}represents the experts’ preferences of the EG2 expert group.

_{1}and D

_{2}is performed. For the uncertainty fusion, the rule for the combination of D numbers ${D}_{ij}={D}_{ij}^{1}\odot {D}_{ij}^{2}$ (Equation (3)) is used. Thus, an aggregated D matrix of experts’ preferences is obtained; see Table 3.

_{1}= {(VH,0.2), (VH;EH,0.35), (EH,0.4)} (where VH is ‘very high’ and EH is ‘extremely high’) and D

_{2}= {(VH,0.25), (VH;EH,0.45), (EH,0.1)}. Table 4 provides an analysis of the data on D numbers whose combination was considered, $D={D}_{C2-C1}^{1}\odot {D}_{C2-C1}^{2}$.

_{C}

_{2−C1}= {(VH,0.350), (EH,0.410)} which is in the first position C2−C1. The remaining values of the aggregated D matrix of experts’ preferences are obtained in a similar way (Table 3).

_{i}) and the total direct/indirect effects that the criterion j receives from other criteria (C

_{i}). The values of R

_{i}and C

_{i}are obtained by using Equations (21) and (22). After calculating the values of R

_{i}and C

_{i}(Table 8), we can obtain the optimal values of the dimensions by using Equations (23)–(26).

_{1}= 0.226), assurance (w

_{2}= 0.211), tangibles (w

_{3}= 0.213), empathy (w

_{4}= 0.160), and responsiveness (w

_{5}= 0.190). Based on presented results, we can define the final ranking as C1 > C3 > C2 > C5 > C4.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Linguistic Variables | Trapezoidal Fuzzy Number |
---|---|

Extremely low (EL) | (0, 1, 2, 3) |

Very low (VL) | (1, 2, 3, 4) |

Low (L) | (2, 3, 4, 5) |

Medium low (ML) | (3, 4, 5, 6) |

Medium (M) | (4, 5, 6, 7) |

Medium high (MH) | (5, 6, 7, 8) |

High (H) | (6, 7, 8, 9) |

Very high (VH) | (7, 8, 9, 10) |

Extremely high (EH) | (8, 9, 10, 10) |

Dim. | C1 | C2 |
---|---|---|

C1 | D_{1} = {(0,0.00)}; | D_{1} = {(H,0.3),(H;VH,0.25),(H,0.4)}; |

D_{2} = {(0,0.00)} | D_{2} = {(VH,0.3),(VH;EH,0.4),(EH,0.3)} | |

C2 | D_{1} = {(VH,0.2),(VH;EH,0.35),(EH,0.4)}; | D_{1} = {(0,0.00)}; |

D_{2} = {(VH,0.25),(VH;EH,0.45),(EH,0.1)} | D_{2} = {(0,0.00)} | |

C3 | D_{1} = {(L,0.2),(ML,0.6),(M,0.15)}; | D_{1} = {(ML,0.2),(ML;M,0.2),(M,0.55)}; |

D_{2} = {(ML,0.35),(ML;M,0.45)} | D_{2} = {(ML;M,0.3),(M,0.3),(M;MH,0.4)} | |

C4 | D_{1} = {(EL,0.4),(EL;VL,0.3),(VL,0.3)}; | D_{1} = {(EL,0.4),(EL;VL,0.4),(VL,0.1)}; |

D_{2} = {(EL;VL,0.25),(VL,0.35),(VL;L,0.35)} | D_{2} = {(EL;VL,0.5),(VL,0.25),(L,0.2)} | |

C5 | D_{1} = {(EL,0.25),(VL,0.55),(VL;L,0.15)}; | D_{1}= {(L,0.4),(ML,0.25),(ML;M,0.35)}; |

D_{2} = {(EL,0.2),(EL;VL,0.5),(VL,0.25)} | D_{2} = {(ML,0.3),(ML;M,0.35),(M,0.3)} | |

C3 | C4 | |

C1 | D_{1} = {(H,0.4),(VH,0.3),(VH;EH,0.3)} | D_{1} = {(MH;H,0.3),(H,0.6),(VH,0.1)} |

D_{2} = {(VH,0.3),(VH;EH,0.3),(EH,0.4)} | D_{2} = {(M,0.3),(MH,0.45),(H,0.25)} | |

C2 | D_{1} = {(MH,0.3),(MH;H,0.35),(H,0.3)} | D_{1} = {(VH,0.3),(VH;EH,0.4),(EH,0.25)} |

D_{2} = {(MH,0.45),(H,0.45)} | D_{2} = {(VH,0.3),(EH,0.4),(H,0.15)} | |

C3 | D_{1} = {(0,0.00)}; | D_{1} = {(MH,0.6),(MH;H,0.2),(H,0.2)} |

D_{2} = {(0,0.00)} | D_{2} = {(M,0.25),(MH,0.35),(H,0.4)} | |

C4 | D_{1} = {(L,0.1),(ML,0.55),(M,0.3)}; | D_{1} = {(0,0.00)}; |

D_{2} = {(VL,0.35),(L,0.45),(ML,0.2)} | D_{2} = {(0,0.00)} | |

C5 | D_{1} = {(MH,0.3),(H,0.25),(H;VH,0.45)}; | D_{1} = {(ML,0.2),(ML;M,0.35),(M,0.4)}; |

D_{2} = {(H,0.3),(H;VH,0.25),(VH,0.45)} | D_{2} = {(ML,0.25),(ML;M,0.3),(M,0.4)} | |

C5 | ||

C1 | D_{1} = {(VH,0.5),(VH;EH,0.1),(EH,0.35)}; | |

D_{2} = {(VH,0.35),(VH;EH,0.2),(EH,0.45)} | ||

C2 | D_{1} = {(MH,0.1),(MH;H,0.3),(H,0.4),(VH,0.15)}; | |

D_{2} = {(MH;H,0.35),(H,0.25),(VH,0.3)} | ||

C3 | D_{1} = {(MH,0.15),(MH;H,0.2),(H,0.55)}; | |

D_{2} = {(MH,0.25),(MH;H,0.35),(H,0.35)} | ||

C4 | D_{1} = {(EL,0.4),(EL;VL,0.4),(VL,0.2)}; | |

D_{2} = {(EL,0.25),(EL;VL,0.35),(VL,0.3)} | ||

C5 | D_{1} = {(0,0.00)}; | |

D_{2} = {(0,0.00)} |

Dim. | C1 | C2 |
---|---|---|

C1 | D = {(0,0.00)} | D = {(VH,0.95)} |

C2 | D = {(VH,0.27),(VH;EH,0.19),(EH,0.3)} | D = {(0,0.00)} |

C3 | D = {(ML,0.67),(M,0.09)} | D = {(ML,0.07),(ML;M5,0.07),(M,0.72)} |

C4 | D = {(EL,0.14),(EL;L,0.11),(2,0.7)} | D = {(EL,0.3),(EL;L,0.3),(L,0.26)} |

C5 | D = {(EL,0.23),(VL,0.68)} | D = {(ML,0.51),(ML;M,0.24),(M,0.2)} |

C3 | C4 | |

C1 | D = {(VH,0.56),(VH;EH,0.19),(EH,0.25)} | D = {(MH,0.38),(H,0.63)} |

C2 | D = {(MH,0.43),(H,0.43)} | D = {(VH,0.36),(EH,0.45)} |

C3 | D = {(0,0.00)} | D = {(MH,0.64),(H,0.36)} |

C4 | D = {(L,0.28),(ML,0.67)} | D = {(0,0.00)} |

C5 | D = {(H,0.46),(H;VH,0.19),(VH,0.34)} | D = {(ML,0.25),(ML;M,0.13),(M,0.52)} |

C5 | ||

C1 | D = {(VH,0.49),(VH;EH,0.03),(EH,0.43)} | |

C2 | D = {(MH,0.06),(MH;H,0.18),(H,0.54),(VH,0.08)} | |

C3 | D = {(MH,0.18),(MH;H,0.09),(H,0.59)} | |

C4 | D = {(EL,0.42),(EL;VL,0.17),(VL,0.31)} | |

C5 | D = {(0,0.00)} |

$\mathit{D}={\mathit{D}}_{\mathit{C}2-\mathit{C}1}^{1}\odot {\mathit{D}}_{\mathit{C}2-\mathit{C}1}^{2}$ | D_{C}_{2−C1}^{2}(VH) = 0.25 | D_{C}_{2−C1}^{2}(VH;EH) = 0.45 | D_{C}_{2−C1}^{2}(EH) = 0.1 |
---|---|---|---|

D_{C}_{2−C1}^{1}(VH) = 0.2 | {VH} (0.05) | {VH} (0.09) | Ø (0.02) |

D_{C}_{2−C1}^{1}(VH;EH) = 0.35 | {VH} (0.0875) | {VH;EH} (0.1575) | {EH} (0.035) |

D_{C}_{2−C1}^{1}(EH) = 0.4 | Ø (0.1) | {EH} (0.18) | {EH} (0.04) |

Dim. | C1 | C2 | C3 |
---|---|---|---|

C1 | (0.00, 0.00, 0.00, 0.00) | (6.65, 7.6, 8.55, 9.5) | (7.36, 8.36, 9.36, 10) |

C2 | (5.73, 6.49, 7.25, 7.6) | (0.00, 0.00, 0.00, 0.00) | (4.7, 5.56, 6.41, 7.27) |

C3 | (2.37, 3.13, 3.89, 4.65) | (3.32, 4.18, 5.03, 5.89) | (0.00, 0.00, 0.00, 0.00) |

C4 | (0.76, 1.71, 2.66, 3.61) | (0.41, 1.26, 2.12, 2.97) | (2.57, 3.52, 4.47, 5.42) |

C5 | (0.68, 1.58, 2.48, 3.38) | (3.17, 4.12, 5.07, 6.02) | (6.44, 7.44, 8.44, 9.44) |

C4 | C5 | ||

C1 | (5.63, 6.63, 7.63, 8.63) | (7.36, 8.31, 9.26, 9.5) | |

C2 | (6.1, 6.91, 7.71, 8.08) | (5.06, 5.91, 6.77, 7.62) | |

C3 | (5.36, 6.36, 7.36, 8.36) | (4.91, 5.76, 6.62, 7.47) | |

C4 | (0.00, 0.00, 0.00, 0.00) | (0.39, 1.29, 2.19, 3.09) | |

C5 | (3.3, 4.2, 5.1, 6.01) | (0.00, 0.00, 0.00, 0.00) |

Dim. | C1 | C2 | C3 |
---|---|---|---|

C1 | (0.00, 0.00, 0.00, 0.00) | (0.25, 0.25, 0.25, 0.25) | (0.27, 0.27, 0.27, 0.27) |

C2 | (0.21, 0.21, 0.21, 0.20) | (0.00, 0.00, 0.00, 0.00) | (0.17, 0.18, 0.18, 0.19) |

C3 | (0.09, 0.10, 0.11, 0.12) | (0.12, 0.14, 0.14, 0.16) | (0.00, 0.00, 0.00, 0.00) |

C4 | (0.03, 0.06, 0.08, 0.10) | (0.02, 0.04, 0.06, 0.08) | (0.10, 0.11, 0.13, 0.14) |

C5 | (0.03, 0.05, 0.07, 0.09) | (0.12, 0.13, 0.15, 0.16) | (0.24, 0.24, 0.24, 0.25) |

C4 | C5 | ||

C1 | (0.21, 0.21, 0.22, 0.23) | (0.27, 0.27, 0.27, 0.25) | |

C2 | (0.23, 0.22, 0.22, 0.21) | (0.19, 0.19, 0.19, 0.20) | |

C3 | (0.20, 0.21, 0.21, 0.22) | (0.18, 0.19, 0.19, 0.20) | |

C4 | (0.00, 0.00, 0.00, 0.00) | (0.01, 0.04, 0.06, 0.08) | |

C5 | (0.12, 0.14, 0.15, 0.16) | (0.00, 0.00, 0.00, 0.00) |

Dim. | C1 | C2 | C3 |
---|---|---|---|

C1 | (0.16, 0.22, 0.28, 0.35) | (0.42, 0.48, 0.54, 0.62) | (0.56, 0.62, 0.67, 0.75) |

C2 | (0.30, 0.35, 0.40, 0.46) | (0.18, 0.23, 0.28, 0.35) | (0.42, 0.48, 0.53, 0.61) |

C3 | (0.17, 0.23, 0.28, 0.36) | (0.23, 0.29, 0.35, 0.43) | (0.20, 0.25, 0.30, 0.38) |

C4 | (0.06, 0.11, 0.17, 0.24) | (0.05, 0.12, 0.19, 0.26) | (0.14, 0.21, 0.28, 0.37) |

C5 | (0.11, 0.18, 0.24, 0.32) | (0.21, 0.27, 0.34, 0.42) | (0.37, 0.43, 0.48, 0.57) |

C4 | C5 | ||

C1 | (0.51, 0.57, 0.63, 0.72) | (0.51, 0.56, 0.61, 0.67) | |

C2 | (0.46, 0.51, 0.56, 0.62) | (0.39, 0.44, 0.49, 0.56) | |

C3 | (0.36, 0.42, 0.48, 0.56) | (0.31, 0.37, 0.42, 0.50) | |

C4 | (0.06, 0.12, 0.17, 0.24) | (0.07, 0.14, 0.21, 0.29) | |

C5 | (0.28, 0.35, 0.41, 0.50) | (0.14, 0.19, 0.25, 0.32) |

Dim. | R_{i} | C_{i} | R_{i} + C_{i} | R_{i} − C_{i} |
---|---|---|---|---|

C1 | (2.16, 2.45, 2.73, 3.11) | (0.80, 1.09, 1.37, 1.73) | (4.84, 0.43, −3.01, −2.17) | (−1.23, 3.84, −2.15, 4.40) |

C2 | (1.75, 2.01, 2.26, 2.61) | (1.10, 1.40, 1.69, 2.09) | (4.70, −0.33, −2.55, −1.72) | (−0.76, 3.70, −1.82, 4.12) |

C3 | (1.28, 1.56, 1.84, 2.23) | (1.69, 1.99, 2.28, 2.69) | (4.92, −1.41, −2.13, −1.27) | (−0.28, 3.85, −1.53, 4.14) |

C4 | (0.38, 0.71, 1.02, 1.40) | (1.68, 1.97, 2.25, 2.64) | (4.05, −2.27, −1.30, −0.42) | (0.59, 2.99, −0.86, 3.11) |

C5 | (1.11, 1.42, 1.73, 2.14) | (1.41, 1.70, 1.98, 2.35) | (4.49, −1.23, −2.01, −1.14) | (−0.18, 3.44, −1.40, 3.71) |

W_{j} | w_{j} | Rank | ||

C1 | 4.404 | 0.226 | 1 | |

C2 | 4.122 | 0.211 | 3 | |

C3 | 4.144 | 0.213 | 2 | |

C4 | 3.110 | 0.160 | 5 | |

C5 | 3.712 | 0.190 | 4 |

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## Share and Cite

**MDPI and ACS Style**

Pribićević, I.; Doljanica, S.; Momčilović, O.; Das, D.K.; Pamučar, D.; Stević, Ž.
Novel Extension of DEMATEL Method by Trapezoidal Fuzzy Numbers and D Numbers for Management of Decision-Making Processes. *Mathematics* **2020**, *8*, 812.
https://doi.org/10.3390/math8050812

**AMA Style**

Pribićević I, Doljanica S, Momčilović O, Das DK, Pamučar D, Stević Ž.
Novel Extension of DEMATEL Method by Trapezoidal Fuzzy Numbers and D Numbers for Management of Decision-Making Processes. *Mathematics*. 2020; 8(5):812.
https://doi.org/10.3390/math8050812

**Chicago/Turabian Style**

Pribićević, Ivan, Suzana Doljanica, Oliver Momčilović, Dillip Kumar Das, Dragan Pamučar, and Željko Stević.
2020. "Novel Extension of DEMATEL Method by Trapezoidal Fuzzy Numbers and D Numbers for Management of Decision-Making Processes" *Mathematics* 8, no. 5: 812.
https://doi.org/10.3390/math8050812