# Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties
## 2 Model Description and Establishment
### Interval Two-Stage Fuzzy Credibility Constraint Programming  
By synthesizing the above models, this paper establishes the interval two-stage fuzzy credibility constraint programming model for optimizing emergency resource scheduling in railway emergencies. The objective function is to achieve the minimum cost under the premise of satisfying the various constraints. The constraints include constraints on the total amount of supplies in the rescue base, pre-decision-making constraints, constraints on the amount of aid-assisted materials, and non-negative constraints. The model is shown as follows:  
The objective function is:  
\[min\ f^{\pm}=\sum_{i=1}^{l}n_{i}k+\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{n=1}^{N}n_{ i}^{\pm}i_{i}^{m}x_{ijn}^{\pm}+\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{h=1}^{H} \sum_{n=1}^{N}P_{h}c_{j}^{\pm}l_{i}^{n}y_{ijlm}^{\pm} \tag{25}\]  
where \(f^{\pm}\) is the system cost; \(A_{n}\) represents the \(n_{th}\) accident demand point; \(R_{i}\) is the \(i_{th}\) emergency rescue base; \(h\) represents the level of material reserves in the rescue base, where \(h=1,2,3\), respectively, refers to three different levels: low, medium and high; \(P_{h}\) represents the probability under the condition \(h\) of the material reserve of the rescue base; \(n_{i}\) represents whether the \(i_{th}\) rescue base provides rescue, \(n_{i}=0,1\), where 1 means providing rescue, 0 means not providing rescue; \(k\) represents the basic start-up costs of the emergency rescue base; \(a_{j}^{\pm}\) represents the unit dispatch cost of the category \(j\) relief supplies; \(c_{j}^{\pm}\) represents the additional penalty cost for other related departments to provide category \(j\) materials; \(l_{i}^{n}\) represents the distance between the emergency rescue base \(R_{i}\) and the accident spot \(A_{n}\). The constraint condition is:  
\[C_{r}\Big{\{}x_{ijn}^{\pm}-y_{ijlm}^{\pm}\leq\widetilde{S}_{ijh}\Big{\}}\geq \lambda_{i} \tag{26}\]  
The above constraint represents the limitations imposed on the total amount of materials available at the rescue base. Where \(x_{ijn}^{\pm}\) represents the amount of type \(j\) materials that are required to be provided by accident spot \(A_{n}\) as the pre-decision rescue point \(i\); \(y_{ijlm}^{\pm}\) represents the number of emergency rescue materials of type \(j\) provided by other relevant departments at the rescue point of \(i\) to the accident spot of \(A_{n}\) under the reserve level \(h\); \(\widetilde{S}_{ijh}\) represents the quantity of emergency rescue materials of category \(j\) at the \(i_{th}\) rescue base in the reserve status \(h\); \(Cr\) is the credibility measure; \(\lambda_{i}\) is the credibility confidence level. \[\sum_{i=1}^{I}x_{ijn}^{\pm}\geq Q_{qj}^{n} \tag{27}\]  
The above constraint represents that the sum of the amount of pre-decision is not less than the number of materials required for emergencies. Where \(Q_{qj}^{n}\) represents the total amount of emergency rescue materials of category \(j\) that are required by accident spot \(A_{n}\) in an emergency with a scenario level of \(q\). \(q\) represents the scale of emergencies, \(q=1,2,3,4\), respectively, refers to four different levels of scenarios from low to high. \[y_{ijhn}^{\pm}>0,\ \sum_{n=1}^{N}x_{ijn}^{\pm}-\sum_{n=1}^{N}y_{ijhn}^{\pm} \geq S_{ijha} \tag{28}\]  
The above constraint indicates that the amount of materials for the auxiliary rescue is provided when the supply is insufficient. Where \(S_{ijha}\), \(S_{ijhb}\) and \(S_{ijhc}\) are all real numbers greater than 0, which are, respectively, the lower limit value, the maximum value, and the upper limit value of \(\widetilde{S}_{ijh}\). \[x_{ijn}^{\pm}\leq S_{ijhc}+S_{rjc} \tag{29}\]  
The above constraint represents the interval range of the pre-decision quantity. Where \(S_{rju},S_{rjb}\) and \(S_{rjc}\) are all real numbers greater than 0, which are, respectively, the lower limit value, the maximum value, and the upper limit value of \(\widetilde{S}_{rj}\).