# Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties

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## Abstract

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## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review

#### 1.3. Focus for This Paper

- In studying the optimization problem of emergency resource dispatching for major railway emergencies under multiple uncertainties, the uncertainties regarding the demand for emergency resources, reserves, and transportation costs are considered comprehensively, and fuzzy variables, probability distributions, and interval numbers are introduced to represent the multiple uncertainty parameters, respectively.
- In this paper, by combining stochastic mathematical programming, interval parametric programming, and fuzzy mathematical programming methods, an interval two-stage fuzzy credibility constraint model is constructed for studying the interactive coupling effects of various uncertain parameters. The model is solved by using the interval interaction algorithm, and empirical analysis is conducted using the example of China Railway Nanchang Group Co., Ltd. to verify that the method and model proposed in this study are scientific and reasonable in practical applications.

## 2. Model Description and Establishment

#### 2.1. Interval Two-Stage Stochastic Programming

#### 2.2. Fuzzy Credibility Constraint Programming

#### 2.3. Interval Two-Stage Fuzzy Credibility Constraint Programming

## 3. Model Application and Solution

#### 3.1. Lower Bound Submodel

#### 3.2. Upper Bound Submodel

## 4. Instance Analysis

#### 4.1. Basic Information

#### 4.2. Formatting of Mathematical Components

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Confidence Level | Level of Satisfaction |
---|---|

$\lambda =1$ | Rescue base reserves fully meet constraints |

$\lambda =0.9$ | Rescue base reserves mostly meet constraints |

$\lambda =0.8$ | Rescue base reserves basically meet constraints |

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{R}}_{6}$ | |
---|---|---|---|---|---|---|

${l}_{i}^{1}$ | 214,684 | 99,756 | 205,955 | 252,863 | 220,267 | 44,637 |

${l}_{i}^{2}$ | 64,315 | 49,790 | 158,363 | 201,934 | 368,307 | 193,807 |

Scenario | ${\mathit{j}}_{1}$ | ${\mathit{j}}_{2}$ | ${\mathit{j}}_{3}$ |
---|---|---|---|

${q}_{1}$ | 85 | 40 | 65 |

${q}_{2}$ | 50 | 25 | 35 |

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

Unit rescue cost | [45,55] | [80,90] | [50,65] |

Unit penalty cost | [20,24] | [30,35] | [24,28] |

**Table 5.**Quantity and probability distribution of ${R}_{i}$ materials in each emergency rescue base (items).

Rescue Base | Reserve Level | ${\mathit{j}}_{1}$ | ${\mathit{j}}_{2}$ | ${\mathit{j}}_{3}$ | Probability |
---|---|---|---|---|---|

${R}_{1}$ | ${h}_{1}$ | [24,27,29] | [12,14,15] | [19,23,24] | 0.3 |

${h}_{2}$ | [30,32,33] | [16,17,18] | [25,27,29] | 0.6 | |

${h}_{3}$ | [34,35,36] | [19,20,21] | [30,31,32] | 0.1 | |

${R}_{2}$ | ${h}_{1}$ | [23,28,29] | [12,13,15] | [18,22,23] | 0.3 |

${h}_{2}$ | [30,31,32] | [16,18,20] | [24,26,28] | 0.6 | |

${h}_{3}$ | [33,34,35] | [21,22,23] | [29,30,31] | 0.1 | |

${R}_{3}$ | ${h}_{1}$ | [24,27,28] | [12,13,15] | [18,22,23] | 0.3 |

${h}_{2}$ | [29,30,32] | [16,18,20] | [24,25,28] | 0.6 | |

${h}_{3}$ | [33,34,35] | [21,22,23] | [29,30,31] | 0.1 | |

${R}_{4}$ | ${h}_{1}$ | [23,27,29] | [12,14,15] | [17,22,24] | 0.3 |

${h}_{2}$ | [30,31,33] | [16,17,18] | [25,27,29] | 0.6 | |

${h}_{3}$ | [34,35,36] | [19,20,21] | [30,31,32] | 0.1 | |

${R}_{5}$ | ${h}_{1}$ | [24,28,29] | [12,13,15] | [18,22,23] | 0.3 |

${h}_{2}$ | [30,31,33] | [16,18,20] | [24,25,28] | 0.6 | |

${h}_{3}$ | [34,35,36] | [21,22,23] | [29,30,31] | 0.1 | |

${R}_{6}$ | ${h}_{1}$ | [23,27,28] | [12,13,15] | [17,21,23] | 0.3 |

${h}_{2}$ | [29,30,32] | [16,18,20] | [24,25,28] | 0.6 | |

${h}_{3}$ | [33,34,35] | [21,22,23] | [29,30,31] | 0.1 |

**Table 6.**Optimal decision-making quantities under different confidence levels in major accidents ${q}_{2}$.

$\mathsf{\lambda}=0.8$ | $\mathsf{\lambda}=0.9$ | $\mathsf{\lambda}=1$ | |||||
---|---|---|---|---|---|---|---|

${A}_{1}$ | ${R}_{i}$ | ${R}_{3}$ | ${R}_{6}$ | ${R}_{3}$ | ${R}_{6}$ | ${R}_{3}$ | ${R}_{6}$ |

${\mathit{j}}_{1}$ | 18 | 32 | 19 | 31 | 19 | 31 | |

${\mathit{j}}_{2}$ | 8 | 17 | 8 | 17 | 8 | 17 | |

${\mathit{j}}_{3}$ | 10 | 25 | 11 | 24 | 11 | 24 | |

${A}_{2}$ | ${R}_{i}$ | ${R}_{1}$ | ${R}_{2}$ | ${R}_{1}$ | ${R}_{2}$ | ${R}_{1}$ | ${R}_{2}$ |

${\mathit{j}}_{1}$ | 20 | 30 | 20 | 30 | 20 | 30 | |

${\mathit{j}}_{2}$ | 9 | 16 | 9 | 16 | 9 | 16 | |

${\mathit{j}}_{3}$ | 11 | 24 | 11 | 24 | 11 | 24 |

Rescue Cost | $\mathsf{\lambda}=1$ | $\mathsf{\lambda}=0.9$ | $\mathsf{\lambda}=0.8$ |
---|---|---|---|

${f}_{opt}^{-}$ | 1,144,838 | 1,144,538 | 1,129,771 |

${f}_{opt}^{+}$ | 1,345,082 | 1,344,722 | 1,326,026 |

**Table 8.**Optimal decision-making quantities under different confidence levels in an extraordinarily serious accident ${q}_{1}$.

$\mathsf{\lambda}=0.8$ | $\mathsf{\lambda}=0.9$ | $\mathsf{\lambda}=1$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ | ${R}_{i}$ | ${R}_{2}$ | ${R}_{3}$ | ${R}_{6}$ | ${R}_{2}$ | ${R}_{3}$ | ${R}_{6}$ | ${R}_{2}$ | ${R}_{3}$ | ${R}_{6}$ |

${\mathit{j}}_{1}$ | 33 | 20 | 32 | 31 | 23 | 31 | 31 | 23 | 31 | |

${\mathit{j}}_{2}$ | 17 | 6 | 17 | 17 | 6 | 17 | 17 | 6 | 17 | |

${\mathit{j}}_{3}$ | 26 | 14 | 25 | 25 | 16 | 24 | 25 | 16 | 24 | |

${A}_{2}$ | ${R}_{i}$ | ${R}_{1}$ | ${R}_{4}$ | ${R}_{5}$ | ${R}_{1}$ | ${R}_{4}$ | ${R}_{5}$ | ${R}_{1}$ | ${R}_{4}$ | ${R}_{5}$ |

${\mathit{j}}_{1}$ | 33 | 31 | 21 | 32 | 31 | 22 | 32 | 31 | 22 | |

${\mathit{j}}_{2}$ | 17 | 17 | 6 | 17 | 17 | 6 | 17 | 17 | 6 | |

${\mathit{j}}_{3}$ | 27 | 26 | 12 | 26 | 25 | 14 | 26 | 24 | 15 |

Rescue Cost | $\mathsf{\lambda}=1$ | $\mathsf{\lambda}=0.9$ | $\mathsf{\lambda}=0.8$ |
---|---|---|---|

${f}_{opt}^{-}$ | 3,316,654 | 3,304,784 | 3,250,665 |

${f}_{opt}^{+}$ | 3,975,919 | 3,960,736 | 3,891,343 |

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**MDPI and ACS Style**

Tang, Z.; Li, W.; Zhou, S.; Sun, J.
Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties. *Appl. Sci.* **2023**, *13*, 4432.
https://doi.org/10.3390/app13074432

**AMA Style**

Tang Z, Li W, Zhou S, Sun J.
Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties. *Applied Sciences*. 2023; 13(7):4432.
https://doi.org/10.3390/app13074432

**Chicago/Turabian Style**

Tang, Zhaoping, Wenda Li, Shengyu Zhou, and Jianping Sun.
2023. "Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties" *Applied Sciences* 13, no. 7: 4432.
https://doi.org/10.3390/app13074432