Vulnerability evaluation of freight railway networks using a heuristic routing and scheduling optimization model

Abstract

Railway network is an integral part of the economy of many countries. Identifying critical network elements can help network executives to take appropriate preventive actions before the occurrence of catastrophic disruptions or to add necessary redundancy to enhance the resilience of the rail network. The criticality of an element or a link is measured by calculating the increased cost or delay when that element is disrupted. In this paper, we proposed a framework for assessing the vulnerability of the freight rail networks by introducing two bi-level models. The first model determines those critical links which have the greatest impact on the routing cost when interdicted, and the next one takes the cost of rescheduling into account, in addition to that of routing, over all origins and destinations. Rerouting effects are already well-captured by existing alternative measures, but this study allows capturing the ramifications of rescheduling measures. The trains are scheduled by a time–space framework considering customer demand, track and station capacities, and time planning horizon. To overcome the difficulty of solving bi-level models, they are converted to single level models. To verify the models and the proposed approach, we considered a case study focused on three disruption scenarios for the railway network of Iran. The accuracy of the obtained results indicates the effectiveness of the proposed methodology. In addition, our method has a very short computational time in comparison with the network scan method (a full enumeration approach).

Appendix: Mathematical proofs

Lemma 1

The optimal solutions of Problem 1 and Problem 2 are equal for a fixed $y$.

Proof

According to Williams (2013), the optimal values of the decision variables of a linear programming problem with an integer right-hand side would be also integer, if the coefficient matrix ($A$) is totally unimodular. A sufficient condition for total unimodularity of a matrix can be defined as property P.

Property P: (from Williams 2013)

• 1. Each element of $A$ is 0, 1 or -1.

• 2. In each column, at most two non-zero elements can appear.

• 3. The rows can be partitioned into two subsets $P_{1}$ and $P_{2}$ such that

  • a. if a column contains two non-zero elements of the same sign, one element is in each of the subsets;

  • b. if a column contains two non-zero elements of opposite sign, both elements are in the same subset.

Problem 2 is a linear programming model, and its coefficients’ matrix holds sufficient condition for total unimodularity that ensures that every basic feasible solution is automatically integer. According to LP theory, one optimal solution exists that is a basic feasible solution, therefore an optimal solution of Problem 2 is integer and satisfies constraint (1 g).$\square$

Theorem 1

The optimal objective function of Problem 3 is equal to the one of the bi-level model (1a)-(1 g).

Proof

It can be proved by contradiction, so suppose that the optimal objective functions are not equal.

Problem 3, according to its definition, is the dual of Problem 2, for any fixed value of $y$. First, we denote $y^{*}$ and $z^{*}$ as the optimal values of $y$, and the optimal value of the objective function in the bi-level model, respectively. Then, if we rewrite Problem 3 with a fixed value $y^{*}$, its optimal objective function would be equal to the optimal objective value of Problem 2, and according to Lemma 1, is equal to $z^{*}$. We also denote $y^{**}$ and $w^{*}$ as the optimal values of $y$ and objective function in Problem 3, respectively. Since $z^{*}$ is a feasible solution for problem 3, $w^{*} \ge z^{*}$. Likewise, the bi-level model can be converted to problem 2 after fixing $y$ at $y^{**}$ and its optimal objective value would be $w^{*}$. As this value, $w^{*}$, is a feasible solution for the bi-level model, it would be less than the optimal value of its objective function, $w^{*} \le z^{*}$. Comparing the last two inequalities, it is concluded that $w^{*} = z^{*}$, the optimal objective values of the bi-level model and problem 3 are the same, completing the proof.$\square$

Linearization procedure

It should be noted that the objective function (2a) contains non-linear terms $u_{ij}^{r} y_{ij}$ which can be linearized by substituting variable $\bar{u}_{ij}^{r} = u_{ij} y_{ij}$ and applying constraint sets (8)-(11).

$$\bar{u}_{ij}^{r} - My_{ij} \le 0$$

(8)

$$- u_{ij}^{r} + \bar{u}_{ij}^{r} \le 0$$

(9)

$$u_{ij}^{r} - \bar{u}_{ij}^{r} + My_{ij} \le M$$

(10)

$$\bar{u}_{ij}^{r} \ge 0$$

(11)

where $M$ is a big number. Finally, the proposed primary model of vulnerability would be:

$$\max_{y \in Y} \quad W(y) = \sum\limits_{r} {(\alpha_{O}^{r} - \alpha_{D}^{r} )} + \sum\limits_{r} {\sum\limits_{i} {\sum\limits_{j} {(u_{ij}^{r} - \bar{u}_{ij}^{r} )} } }$$

(12)

subject to (1b), (1f), (2b-2d) and (3a-3d)