# Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The PIOGPA operator introduces the probability parameter and priority and can reflect the importance degrees and priority relationships of attributes;
- (2)
- The PIOGPA operator is more flexible and robust. It considers the priority between different attributes and can manage the influence of extreme data or biased data and obtain more reasonable decision results;
- (3)
- The PIOGPA operator is more applicable in addressing uncertain problems. It can adjust the critical attributes in time and determine the new critical attribute weight.

## 2. Basic Concepts

**Definition**

**1**

**.**Let $m$ be a set of positive integers. Then, the interval ordering $X$ can be expressed as:

**Definition**

**2**

**.**Let $m$ be a set of positive integers. Then, the probabilistic interval ordering set ${H}_{p}$ is given as:

**Definition**

**3**

**.**Let $E\left({\dot{h}}_{p}(m)\right)$ and $S\left({\dot{h}}_{p}(m)\right)$ be the expected and scoring values of the PIPOE ${\dot{h}}_{p}(m)$, respectively, then $E\left({\dot{h}}_{p}(m)\right)$ and $S\left({\dot{h}}_{p}(m)\right)$ can be defined as:

**Definition**

**4**

**.**Let $S\left({\dot{h}}_{p}{\left(m\right)}_{1}\right)$ and $S\left({\dot{h}}_{p}{\left(m\right)}_{2}\right)$ be the scores of two different schemes in the probabilistic interval ordering set; then:

- (1)
- when $P\left(S\left({\dot{h}}_{p}{\left(m\right)}_{1}\right)>S\left({\dot{h}}_{p}{\left(m\right)}_{2}\right)\right)<0.5$, ${\dot{h}}_{p}{(m)}_{1}\succ {\dot{h}}_{p}{(m)}_{2}$;
- (2)
- when $P\left(S\left({\dot{h}}_{p}{\left(m\right)}_{1}\right)>S\left({\dot{h}}_{p}{\left(m\right)}_{2}\right)\right)=0.5$, ${\dot{h}}_{p}{(m)}_{1}={\dot{h}}_{p}{(m)}_{2}$;
- (3)
- when $P\left(S\left({\dot{h}}_{p}{\left(m\right)}_{1}\right)>S\left({\dot{h}}_{p}{\left(m\right)}_{2}\right)\right)>0.5$, ${\dot{h}}_{p}{(m)}_{1}\prec {\dot{h}}_{p}{(m)}_{2}$.

**Definition**

**5**

**.**Given a set of attributes $A=\left\{{A}_{1},{A}_{2},\cdots ,{A}_{n}\right\}$, each attribute ${A}_{i}$ contains a set of probabilistic interval orderings ${H}_{p}$ with the importance ranking ${A}_{1}\succ {A}_{2}\succ \cdots \succ {A}_{n}$, and then the attributes are called prioritized attributes. The prioritized averaging operator (PA) is defined as:

**Definition**

**6**

**.**Given a set of attributes $A=\{{A}_{1},{A}_{2},\dots ,{A}_{n}\}$, each attribute ${A}_{i}$ contains a set of probabilistic interval orderings ${H}_{p}$. For convenience, ${h}_{i}\left(i=1,2,\cdots ,n\right)$ is used to represent the ${\dot{h}}_{p}{(m)}_{i}\left(i=1,2,\cdots ,n\right)$. In traditional MADM models, the weight of each attribute follows the assumption of uniform distribution of attribute values. Then, the PIPOWA operator can be expressed as:

## 3. Probabilistic Interval Ordering Averaging Operator with Attribute Priority

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

## 4. MADM Method Based on the PIOGPA Operator

#### 4.1. Consistency Algorithm

- (1)
- when $P\left({\dot{h}}_{p}(m)\ge \mu -3\delta \right)<0.5$, ${\dot{h}}_{p}(m)\succ \mu -3\delta $, ${\dot{h}}_{p}(m)$ is better than $\mu -3\delta $;
- (2)
- when $P\left({\dot{h}}_{p}(m)\ge \mu -3\delta \right)=0.5$, ${\dot{h}}_{p}(m)=\mu -3\delta $, ${\dot{h}}_{p}(m)$ is as good as $\mu -3\delta $;
- (3)
- when $P\left({\dot{h}}_{p}(m)\ge \mu -3\delta \right)>0.5$, ${\dot{h}}_{p}(m)\prec \mu -3\delta $, ${\dot{h}}_{p}(m)$ is not as good as $\mu -3\delta $.

#### 4.2. MADM Steps Based on the PIOGPA Operator

## 5. Case Study

#### 5.1. Case Analysis

#### 5.2. Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

C_{1} | [1,2]_{0.6}, [2,3]_{0.3}, [3,4]_{0.1} | [2,3]_{0.7}, [3,4]_{0.1}, [5,6]_{0.2} | [1,3]_{0.5}, [3,4]_{0.3}, [4,5]_{0.2} | [1,2]_{0.7}, [3,4]_{0.1}, [4,5]_{0.2} |

C_{2} | [1,3]_{0.4}, [3,4]_{0.4}, [5,6]_{0.2} | [1,2]_{0.35}, [2,3]_{0.65} | [2,3]_{0.4}, [3,4]_{0.55}, [4,5]_{0.05} | [1,3]_{0.55}, [3,4]_{0.45} |

C_{3} | [1,2]_{0.2}, [2,3]_{0.8} | [2,3]_{0.9,} [3,4]_{0.09}, [4,5]_{0.01} | [3,4]_{0.9}, [4,5]_{0.1} | [1,2]_{0.6}, [2,3]_{0.3}, [5,6]_{0.1} |

C_{4} | [1,2]_{0.3}, [3,4]_{0.6}, [4,5]_{0.1} | [1,2]_{0.3}, [3,4]_{0.7} | [1,2]_{0.55}, [3,4]_{0.41}, [4,5]_{0.04} | [1,2]_{0.7}, [2,3]_{0.3} |

C_{5} | [1,3]_{0.9}, [3,4]_{0.08}, [5,6]_{0.02} | [2,3]_{0.7}, [3,4]_{0.2}, [5,6]_{0.1} | [1,2]_{0.6}, [3,4]_{0.25}, [5,6]_{0.15} | [1,2]_{0.3}, [2,3]_{0.5}, [4,5]_{0.2} |

C_{6} | [1,2]_{0.75}, [3,4]_{0.25} | [1,2]_{0.6}, [2,3]_{0.3}, [3,4]_{0.1} | [1,2]_{0.8}, [3,5]_{0.17}, [5,6]_{0.03} | [1,2]_{0.6}, [2,4]_{0.4} |

A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

C_{1} | [1,2]_{0.6}, [2,3]_{0.3}, [3,4]_{0.1} | [2,3]_{0.7}, [3,4]_{0.1}, [5,6]_{0.2} | [1,3]_{0.5}, [3,4]_{0.3}, [4,5]_{0.2} | [1,2]_{0.7}, [3,4]_{0.1}, [4,5]_{0.2} |

C_{2} | [1,3]_{0.4}, [3,4]_{0.4}, [5,6]_{0.2} | [1,2]_{0.35}, [2,3]_{0.65} | [2,3]_{0.4}, [3,4]_{0.55}, [4,5]_{0.05} | [1,3]_{0.55}, [3,4]_{0.45} |

C_{3} | [1,2]_{0.2}, [2,3]_{0.8} | [2,3]_{0.9}, [3,4]_{0.09}, [4,5]_{0.01} | [3,4]_{0.9}, [4,5]_{0.1} | [1,2]_{0.6}, [2,3]_{0.3}, [5,6]_{0.1} |

C_{4} | [1,2]_{0.3}, [3,4]_{0.6}, [4,5]_{0.1} | [1,2]_{0.3}, [3,4]_{0.7} | [1,2]_{0.55}, [3,4]_{0.41}, [4,5]_{0.04} | [1,2]_{0.7}, [2,3]_{0.3} |

C_{5} | [1,3]_{0.9}, [3,4]_{0.08}, [5,6]_{0.02} | [2,3]_{0.7}, [3,4]_{0.2}, [5,6]_{0.1} | [1,2]_{0.6}, [3,4]_{0.25}, [5,6]_{0.15} | [1,2]_{0.3}, [2,3]_{0.5}, [4,5]_{0.2} |

C_{6} | [1,2]_{0.75}, [3,4]_{0.25} | [1,2]_{0.6}, [2,3]_{0.3}, [3,4]_{0.1} | [1,2]_{0.8}, [3,5]_{0.17}, [5,6]_{0.03} | [1,2]_{0.6}, [2,4]_{0.4} |

Integrated Operator | The Priority Vector | Sorting Results |
---|---|---|

PIPOA [36] | $\omega ={(0.1870,0.1309,0.1271,0.1706,0.1914,0.1930)}^{T}$ | ${C}_{6}\prec {C}_{5}\prec {C}_{1}\prec {C}_{4}\prec {C}_{2}\prec {C}_{3}$ |

PIPOWA [36] | $\omega ={(0.1858,0.1337,0.1220,0.1691,0.1912,0.1981)}^{T}$ | ${C}_{6}\prec {C}_{5}\prec {C}_{1}\prec {C}_{4}\prec {C}_{2}\prec {C}_{3}$ |

IHFOA | $\omega ={(0.1552,0.1523,0.1435,0.1540,0.1891,0.2059)}^{T}$ | ${C}_{6}\prec {C}_{5}\prec {C}_{1}\prec {C}_{4}\prec {C}_{2}\prec {C}_{3}$ |

IHFOPA | $\omega ={(0.1579,0.1274,0.1075,0.1736,0.2100,0.2236)}^{T}$ | ${C}_{6}\prec {C}_{5}\prec {C}_{4}\prec {C}_{1}\prec {C}_{2}\prec {C}_{3}$ |

IHFOGPA | $\omega ={(0.1567,0.1261,0.1067,0.1740,0.2119,0.2246)}^{T}$ | ${C}_{6}\prec {C}_{5}\prec {C}_{4}\prec {C}_{1}\prec {C}_{2}\prec {C}_{3}$ |

PIOPA | $\omega ={(0.1675,0.1661,0.1007,0.1844,0.1711,0.2101)}^{T}$ | ${C}_{6}\prec {C}_{4}\prec {C}_{5}\prec {C}_{1}\prec {C}_{2}\prec {C}_{3}$ |

PIOGPA | $\omega ={(0.1674,0.1665,0.0994,0.1852,0.1702,0.2114)}^{T}$ | ${C}_{6}\prec {C}_{4}\prec {C}_{5}\prec {C}_{1}\prec {C}_{2}\prec {C}_{3}$ |

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**MDPI and ACS Style**

Ruan, C.; Gong, S.; Chen, X.
Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making. *Axioms* **2023**, *12*, 1007.
https://doi.org/10.3390/axioms12111007

**AMA Style**

Ruan C, Gong S, Chen X.
Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making. *Axioms*. 2023; 12(11):1007.
https://doi.org/10.3390/axioms12111007

**Chicago/Turabian Style**

Ruan, Chuanyang, Shicheng Gong, and Xiangjing Chen.
2023. "Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making" *Axioms* 12, no. 11: 1007.
https://doi.org/10.3390/axioms12111007