# An Inner Dependence Analysis Dynamic Decision-Making Framework

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Model Formulation

#### 3.1. A Dynamic Decision Framework

#### 3.2. Inner Dependence in a Pair-Wise Comparison Matrix

## 4. Case Illustration

#### 4.1. Case 1: The Independence AHP

#### 4.2. Case 2: The All Inner Dependence AHP

#### 4.3. Case 3: The Partial Inner Dependence AHP

_{2}is as follows:

## 5. A Numerical Illustration

- We first interviewed decision maker(s) to estimate rankings of alternatives. The rankings are shown in Table 9. A good assessment of the alternatives relies on the criteria that should influence the selection of alternatives for responding to the goal. The weights of alternatives are estimated by the holistic judgments of the decision maker(s).
- Inter-dependencies among criteria expressed by a pair-wise matrix must also be examined; and the influence of each criterion on each of other criteria can be represented by an eigenvector. Like normal AHP, the pair-wise comparison in inner dependence AHP is performed in a matrix, and a local priority vector can be derived as an estimate of the relative importance associated with the criteria being compared by using Equation (7).
- Relationship (FW = nW)

$$\left[\begin{array}{c}0.314\\ 0.314\\ 0.091\\ 0.166\\ 0.114\end{array}\right]\times \left[\begin{array}{ccccc}1& 1& 3& 2& 3\\ & 1& 3& 2& 3\\ & & 1& 1/3& 1\\ & & & 1& 1\\ & & & & 1\end{array}\right]=5.11\times \left[\begin{array}{c}0.314\\ 0.314\\ 0.091\\ 0.166\\ 0.114\end{array}\right]$$- b.
- Price (FW = nW)

$$\left[\begin{array}{c}0.2\\ 0.2\\ 0.2\\ 0.2\\ 0.2\end{array}\right]\times \left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ & 1& 1& 1& 1\\ & & 1& 1& 1\\ & & & 1& 1\\ & & & & 1\end{array}\right]=5\times \left[\begin{array}{c}0.2\\ 0.2\\ 0.2\\ 0.2\\ 0.2\end{array}\right]$$- c.
- Location (FW = nW)

$$\left[\begin{array}{c}0.2\\ 0.2\\ 0.2\\ 0.2\\ 0.2\end{array}\right]\times \left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ & 1& 1& 1& 1\\ & & 1& 1& 1\\ & & & 1& 1\\ & & & & 1\end{array}\right]=5\times \left[\begin{array}{c}0.2\\ 0.2\\ 0.2\\ 0.2\\ 0.2\end{array}\right]$$ - In Equation (7), where W is the equivalent priority vector of F. ${W}^{T}=\left[{w}_{1},{w}_{2},\dots .,{w}_{n}\right]$ is the transformation matrix of W. ${W}^{T}$ verifying the condition ${a}_{ij}=\frac{{w}_{i}}{{w}_{j}}.$ The consistency condition is given by ${f}_{ik}={f}_{ij}{f}_{jk}\forall i,j,k=1,\dots .,n$. If ${\lambda}_{max}$ is the maximum eigenvalue of F and F is consistent then ${\lambda}_{max}\ge n$, where n is the number of criteria used in the pair-wise comparison matrix. The transformation matrix is used to present the inner dependency among criteria. If the values of the eigenvalue make two matrices equal, the inner dependency exists among criteria, e.g., the dependence conditions in a and the independence conditions in b.
- The dependence condition is expressed below.

$$\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ & 1& 1& 1& 1\\ & & 1& 1& 1\\ & & & 1& 1\\ & & & & 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 5\\ 5\\ 5\\ 5\end{array}\right]$$- The independence condition is expressed below.

$$\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ & 1& 1& 1& 1\\ & & 1& 1& 1\\ & & & 1& 1\\ & & & & 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]\ne \left[\begin{array}{c}5\\ 5\\ 5\\ 5\\ 5\end{array}\right]$$

- (1)
- The AHP evaluates the criteria for decision-making quantitatively;
- (2)
- The AHP is easy to analyze the inner dependence;
- (3)
- The inner dependence reflects the better relationship among criteria; and
- (4)
- The transformation matrix uses one or more numbers, or coordinates, to uniquely determine the position of the points. We used it to present the inner dependence among criteria/matrices.

## 6. Discussions

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Criterion 1 | Criterion 2 | Criterion 3 | Geometric Mean | Eigenvector | |
---|---|---|---|---|---|

Criterion 1 | 1 | 3 | $\frac{1}{3}$ | 1.000 | 1.000/3.871 = 0.258 |

Criterion 2 | $\frac{1}{3}$ | 1 | $\frac{1}{5}$ | 0.405 | 0.405/3.871 = 0.105 |

Criterion 3 | 3 | 5 | 1 | 2.466 | 2.466/3.871 = 0.637 |

Sum | 3.871 |

Criterion 1 | Criterion 2 | Criterion 3 | Geometric Mean | Eigenvector | |
---|---|---|---|---|---|

Criterion 1 | 1 | 3 | 5 | 2.466 | 2.466/3.867 = 0.637 |

Criterion 2 | $\frac{1}{3}$ | 1 | 2 | 0.997 | 0.997/3.867 = 0.258 |

Criterion 3 | $\frac{1}{5}$ | $\frac{1}{2}$ | 1 | 0.404 | 0.404/3.867 = 0.105 |

Sum | 3.867 | ${W}_{1}^{T}=\left[0.637,0.258,0.105\right]$ |

Criterion 1 | Criterion 1 | Criterion 2 | Criterion 3 | Geometric Mean | Eigenvector |
---|---|---|---|---|---|

Criterion 1 | 1 | 5 | 7 | 3.270 | 3.270/4.419 = 0.740 |

Criterion 2 | $\frac{1}{5}$ | 1 | 2 | 0.737 | 0.737/4.419 = 0.166 |

Criterion 3 | $\frac{1}{7}$ | $\frac{1}{2}$ | 1 | 0.412 | 0.412/4.419 = 0.094 |

Sum | 4.419 | ${W}_{2}^{T}=\left[0.740,0.166,0.094\right]$ |

Criterion 2 | Criterion 1 | Criterion 2 | Criterion 3 | Geometric Mean | Eigenvector |
---|---|---|---|---|---|

Criterion 1 | 1 | $\frac{1}{3}$ | 2 | 0.87 | 0.87/3.8 = 0.230 |

Criterion 2 | $3$ | 1 | 5 | 2.47 | 2.47/3.8 = 0.648 |

Criterion 3 | $\frac{1}{2}$ | $\frac{1}{5}$ | 1 | 0.46 | 0.46/3.8 = 0.122 |

Sum | 3.80 | ${W}_{3}^{T}=\left[0.230,0.648,0.122\right]$ |

Criterion 3 | Criterion 1 | Criterion 2 | Criterion 3 | Geometric Mean | Eigenvector |
---|---|---|---|---|---|

Criterion 1 | 1 | $2$ | $\frac{1}{2}$ | 1.0 | 1.0/3.5 = 0.286 |

Criterion 2 | $\frac{1}{2}$ | 1 | $\frac{1}{4}$ | 0.5 | 0.5/3.5 = 0.143 |

Criterion 3 | $2$ | $4$ | 1 | 2.0 | 2.0/3.5 = 0.571 |

Sum | 3.5 | ${W}_{4}^{T}=\left[0.286,0.143,0.571\right]$ |

C1 | C2 | C3 | C4 | C5 | Geometric Mean | Eigenvector | |
---|---|---|---|---|---|---|---|

C1 | 1 | 3 | 2 | 5 | 7 | 2.91 | 2.91/6.68 = 0.44 |

C2 | $\frac{1}{3}$ | 1 | $\frac{1}{2}$ | 2 | 3 | 1.00 | 1.00/6.68 = 0.15 |

C3 | $\frac{1}{2}$ | 2 | 1 | 4 | 6 | 1.89 | 1.89/6.68 = 0.28 |

C4 | $\frac{1}{5}$ | $\frac{1}{2}$ | $\frac{1}{4}$ | 1 | 2 | 0.55 | 0.55/6.68 = 0.08 |

C5 | $\frac{1}{7}$ | $\frac{1}{3}$ | $\frac{1}{6}$ | $\frac{1}{2}$ | 1 | 0.33 | 0.33/6.68 = 0.05 |

C1 | C2 | C3 | C4 | Geometric Mean | Eigenvector | |
---|---|---|---|---|---|---|

C1 | 1 | $\frac{1}{6}$ | 2 | $\frac{1}{2}$ | 0.64 | 0.640/5.82 = 0.110 |

C2 | 6 | 1 | 9 | 3 | 3.57 | 0.737/5.82 = 0.613 |

C3 | $\frac{1}{2}$ | $\frac{1}{9}$ | 1 | $\frac{1}{4}$ | 0.34 | 0.412/5.82 = 0.058 |

C4 | 2 | $\frac{1}{3}$ | 4 | 1 | 1.27 | 1.270/5.82 = 0.218 |

C3 | C4 | Geometric Mean | Eigenvector | |
---|---|---|---|---|

C3 | 1 | $\frac{1}{5}$ | 0.45 | 0.45/2.69 = 0.167 |

C4 | 5 | 1 | 2.24 | 2.24/2.69 = 0.832 |

C1 | C2 | C3 | C4 | C5 | Weight | Ranking | |
---|---|---|---|---|---|---|---|

C1 | 1 | 2 | 4 | 2 | 2 | 2.000 | 1 |

C2 | $\frac{1}{2}$ | 1 | 3 | 1 | 2 | 1.246 | 2 |

C3 | $\frac{1}{4}$ | $\frac{1}{3}$ | 1 | $\frac{1}{2}$ | $\frac{1}{3}$ | 0.425 | 5 |

C4 | $\frac{1}{2}$ | 1 | 2 | 1 | 1 | 1.000 | 3 |

C5 | $\frac{1}{2}$ | $\frac{1}{2}$ | 3 | 1 | 1 | 0.944 | 4 |

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**MDPI and ACS Style**

Liu, Y.-N.; Wu, H.-H.
An Inner Dependence Analysis Dynamic Decision-Making Framework. *Sustainability* **2022**, *14*, 5968.
https://doi.org/10.3390/su14105968

**AMA Style**

Liu Y-N, Wu H-H.
An Inner Dependence Analysis Dynamic Decision-Making Framework. *Sustainability*. 2022; 14(10):5968.
https://doi.org/10.3390/su14105968

**Chicago/Turabian Style**

Liu, Yun-Ning, and Hsin-Hung Wu.
2022. "An Inner Dependence Analysis Dynamic Decision-Making Framework" *Sustainability* 14, no. 10: 5968.
https://doi.org/10.3390/su14105968