Train timetabling at rapid rail transit lines: a robust multi-objective stochastic programming approach

Abstract

In the setting of public transportation system, improving the service quality as well as robustness against uncertainty through minimizing the total waiting times of passengers is a real issue. This study proposed robust multi-objective stochastic programming models for train timetabling problem in urban rail transit lines. The objective is to minimize the expected value of the passenger waiting times, its variance and the penalty cost function including the capacity violation due to overcrowding. In the proposed formulations, the dynamic and uncertain travel demand is represented by the scenario-based time-varying arrival rates and alighting ratio at stops. Two versions of the robust stochastic programming models are developed and a comparative analysis is conducted to testify the tractability of the models. The effectiveness of the proposed stochastic programming model is demonstrated through the application to line 5 of Tehran underground railway. The outcomes validate the benefits of implementing robust timetables for rail industry. The computational experiments shows significant reductions in expected passenger waiting time of 21.27 %, and cost variance drop of 59.98 % for the passengers, through the proposed robust mathematical modeling approach.

Appendices

Appendix 1

Lemma 1

Consider a goal programming model as follows:

$$Minimize\quad Z = \left| {f\left( X \right) - g} \right|$$

$$\begin{aligned}& {subject\, to:}\\&X \in F \left( {F\; is\; a\; feasible\; region} \right) \end{aligned}$$

The objective function can be linearized using the following way:

$$Minimize\quad Z^{\prime} = f\left( X \right) - g + 2\tau$$

$$\begin{aligned}& {subject\, to{:}}\\& g - f\left( X \right) + \tau \le 0\end{aligned}$$

$$\tau \ge 0, \;X \in F$$

Proof

In order to prove the above lemma, two cases are investigated.

$${\text{CASE}}\;{1{:}}\;f\left( X \right) - g \ge 0.$$

In this case, at the optimal condition, τ is required as τ = 0, which results in Z′ = Z.

$${\text{CASE}}\;{2{:}}\;f\left( X \right) - g < 0.$$

In this case, at the optimal condition, τ is required as τ = g − f(X), which results in Z′ = f(X) − g + 2τ = g − f(X) = Z.

Appendix 2: Stochastic (Poisson) arrival process

In this section, we prove that the expected waiting time is equivalent for both deterministic and stochastic arrival processes (Poisson arrival process). The order-statistic property of Poisson process can be used here to show it directly (Ross 1996). Let {N(t)} denotes the arrival process of passengers. Suppose N(t) = n passenger arrival times t ₁ , …, t _n as the order statistics U ₍₁₎  < U ₍₂₎  < ⋯ < U _(n) of n independent uniform placed in ascending order during the interval [0, T] are given. Therefore, U ₍₁₎ is the smallest one, U ₍₂₎ the second smallest and lastly U _(n) is the largest value. U _(i) is called the i-th order statistic of U ₁ ,…,U _n . The average waiting time of a passenger is denoted by $E\left( w \right)$. The ith arrival has waiting time $t - t_{i}$. If there will be N(t) such arrivals, then the waiting time of a passenger is calculated as:

$$w = \frac{1}{N\left( t \right)}\mathop \sum \limits_{i = 1}^{N\left( t \right) } \left( {t - t_{i} } \right)$$

(78)

When $N\left( t \right) = n$ is given, we can drive the following equation simply because the sum of all n of $U_{\left( i \right)}$ is the same as the sum of all n of the $U_{i}$.

$$w = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {t - U_{\left( i \right)} } \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {t - U_{i} } \right) \to E\left( {w |N\left( t \right) = n} \right) = \frac{1}{n}nE\left( {t - U} \right) = E\left( {t - U} \right) = \frac{t}{2}$$

(79)

The above equation is hold for all $n \ge 1$, we get $E\left( w \right) = t/2$. As a result, it is expected that on average a passenger waits $E\left( w \right) = t/2$ units of time. The average number of passengers arriving during $\left[ {0,T} \right]$ is $\lambda t$. Accordingly, the expected waiting time of all passengers equals to $\lambda t*\frac{t}{2} = \frac{1}{2}\lambda t^{2}$. As a result, the expected value of the waiting times remains valid whether trains arrive regularly spaced or arrive according to the Poisson process are assumed.