Exact and heuristic approaches to the robust periodic event scheduling problem

Abstract

In the periodic event scheduling problem, periodically reoccurring events need to be scheduled, subject to constraints on the resulting time differences. A typical application for this type of problem relates to train schedules, which have to repeat every hour for passenger convenience. As external disruptions may occur, robustness considerations need to be included in the scheduling process. In this work, we present a recovery approach for instances where integer programming methods can be applied, and a bi-criteria local search algorithm for large-scale instances. In computational experiments, we compare solutions calculated using the recovery approach to risk-averse and to risk-oblivious solutions. Our results suggest that the solutions generated by our approach have a favorable trade-off between cost and robustness. Furthermore, we compare the local search algorithm to a simplified approach that includes the desired robustness level as a hard constraint. The experiments show that our algorithm finds an improved set of non-dominated solutions within equal computation times.

Appendix: Decomposition of RecOpt-PTT

The following lemma is given in Goerigk and Schöbel (2014).

Lemma 1

Let $x^1, \ldots , x^N\in \mathbb {R}^n$ be optimal solutions to the scenarios $\xi ^1,\ldots ,\xi ^N$ and let $x^*$ be the solution of a location problem (Loc) that is of one of the following forms:

• 1. $loc(x)= \sum _{i=1}^N d^2(x^i,x)$, where $d^2$ is the squared Euclidean distance.

• 2. $loc(x)= \sum _{i=1}^N d(x^i,x)$, where $d$ is linear equivalent to the Euclidean distance.

• 3. $loc(x)= \max _{i=1}^N d(x^i,x)$, where $d$ is linear equivalent to the Euclidean distance.

• 4. $loc(x)= \sum _{i=1}^N d(x^i,x)$, where $d$ is any $l_p$-norm for $1 < p < \infty$ and $n=2$.

Let $Q(x^i) \le \delta$, $i=1,\ldots ,N$, for some convex function $Q:\mathbb {R}^n \rightarrow \mathbb {R}$, $\delta \in \mathbb {R}$. Then

$$\begin{aligned} Q(x^*) \le \delta . \end{aligned}$$

Definition 1

Let $P$ be an optimization problem, and let $(\mathcal {X}_1,\mathcal {X}_2)$ be a partition of the problem variables. Let $\mathcal {F}$ be the set of feasible solutions for $P$. We call $P$ combinable with respect to the variables $\mathcal {X}_1$, if

$$\begin{aligned} x_1 \in Pr_{\mathcal {X}_1} (\mathcal {F}) \end{aligned}$$

for all $x_1 \in \mathrm{conv} \{x^1_1, \ldots , x^N_1\}$ and every set of feasible solutions $\{(x^1_1,x^1_2), \ldots ,(x^N_1,x^N_2)\}$ to $P$.

Theorem 1

Let $(P(\xi ),\xi \in \mathcal {U})$ be an uncertain optimization problem with finite uncertainty, and let $P(\hat{\xi })$ be combinable with respect to the variables $\mathcal {X}_1$. Let $\mathcal {F}(\xi ) \subseteq \mathcal {F}(\hat{\xi })$ for all $\xi \in \mathcal {U}$. Let $d$ be a metric of any of the types from Lemma 1 that only depends on the variables $\mathcal {X}_1$. Then there is a solution to (RecOpt) that is feasible for the nominal scenario.

Proof

Let $\{x^1,\ldots ,x^N\}$ be optimal solutions to the scenarios $\{\xi ^1,\ldots ,\xi ^N\}$, respectively. Let $(x_1^*, x_2^*)\in \mathcal {X}_1\times \mathcal {X}_2$ be an optimal solution to the location problem (Loc). Due to Lemma 1, we have $x_1^* \in \mathrm{conv}\{x^1_1, \ldots , x^N_1\}$.

As $P$ is combinable, there is an $x_2'$ such that $(x_1^*,x_2')$ is feasible for the nominal problem. As $d$ does not depend on the variables in $\mathcal {X}_1$, $(x_1^*,x_2')$ has the same objective value for (Loc) as $(x_1^*, x_2^*)$, which completes the proof.

We conclude that in the case of periodic timetabling, we can separately solve every scenario, solve an unconstrained location problem, and find an optimal solution to (RecOpt-PTT):

Corollary 1

If there is a unique optimal solution to every scenario, (RecOpt-PTT) with respect to a metric equivalent to $l_2$, or $l_2^2$ considering only activities from $\mathcal {A}_{drive} \cup \mathcal {A}_{wait}$ can be solved to optimality by solving $N$ periodic timetabling problems, and one unconstrained location problem.