# Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties
## 2 Model Description and Establishment  
### Interval Two-Stage Stochastic Programming  
The interval two-stage stochastic programming method is a combination of interval parameter programming and two-stage stochastic programming. Both methods have unique advantages in solving problems with uncertain variables. The advantage of two-stage stochastic programming is to optimize the decision-making process under the probability distribution representation of different scenarios. However, the advantage of interval parameter programming is to express the uncertainty of discrete intervals and ensure that the solution interval covers the true value in the form of intervals. The general form of two-stage stochastic planning is:  
\[\text{min}f=cx+E[Q(y,\delta)] \tag{1}\]where \(x\) is the first-stage decision made before the occurrence of the stochastic scenario; \(y\) is the second-stage adaptive decision, which depends on the realization of the random variable \(\delta\); \(E\) denotes the mathematical expectation function; \(Q(y,\delta)\) denotes the cost function of the second stage; \(\delta\) is the decision variable of the second stage, and it is also a random variable whose probability distribution is difficult to determine. Considering the actual situation, the resource reserves of the emergency rescue base are treated as discrete variables, and the probability of the corresponding changes in reserves is \(P_{h}\). The expression is as follows:  
\[\sum_{h=1}^{H}P_{h}=1,\ h=1,2,\ldots,H \tag{2}\]  
\[E[Q(y,\delta)]=EQ(y)=\sum_{h=1}^{H}P_{h}Q(y,\delta_{h}) \tag{3}\]  
where \(h\) represents the level of material reserves in the rescue base, \(h=1\) indicates the lowest level, and \(h=H\) indicates the highest level; \(P_{h}\) represents the probability under the condition \(h\) of the material reserve of the rescue base; \(\delta_{h}\) represents the random variable corresponding to a material reserve level of \(h\); each \(\delta_{h}\) corresponds to a different second-stage decision, denoted by \(y_{h}\). The linear form of the model is expressed as:  
\[\text{min}f=cx+\sum_{h=1}^{H}P_{h}Q(y_{h},\delta_{h}) \tag{4}\]  
The constraint condition is shown as follows:  
\[Ax\leq b \tag{5}\]  
\[x\geq 0 \tag{6}\]  
\[y_{h}\geq 0 \tag{7}\]  
\[T(\delta_{h})x+W(\delta_{h})y_{h}=g(\delta_{h})\quad\forall h=1,2,\ldots,H \tag{8}\]  
where \(A\) is the constant coefficient matrix of the first-stage decision variable \(x\); \(b\) is a constant matrix; \(T(\delta_{h})\), \(W(\delta_{h})\), and \(g(\delta_{h})\) are the stochastic model parameters with reasonable dimensions, which are functions of the random variable \(\delta_{h}\). The decision variables of the system are usually represented as interval parameters with upper and lower bounds; therefore, the interval parameters are introduced into the model to obtain the interval two-stage stochastic planning model (ITSP), which is represented as follows:  
\[\text{min}f^{\pm}=c^{\pm}x^{\pm}+\sum_{h=1}^{H}P_{h}Q(y^{\pm},\delta_{h}) \tag{9}\]  
where \(x\) is a closed and bounded set of real numbers; \(x^{\pm}\) is an interval with known upper and lower bounds, while the distribution information for \(x\) is unknown. The constraint condition is shown as follows:  
\[A^{\pm}x^{\pm}\leq b^{\pm} \tag{10}\]  
\[\sum_{h=1}^{H}P_{h}=1,\ h=1,2,\ldots,H \tag{2}\]0  
\[\sum_{h=1}^{H}P_{h}=1,\ h=1,2,\ldots,H \tag{2}\]1\[\sum_{h=1}^{H}P_{h}=1,\ h=1,2,\ldots,H \tag{2}\]2  
In research on the optimization of emergency resource scheduling for railway emergencies, full consideration should be given to the various railway emergencies that may occur in the future and the influence of the uncertain variables in them. The railway emergency scheduling model, constructed through interval two-stage programming, can scientifically and effectively respond to the various scenarios of railway emergencies that may occur in the future, thus providing feasible and reliable solutions.