Marc Goerigk

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions.

An empirical analysis of robustness concepts for timetabling

Calculating timetables that are insensitive to disturbances has drawn considerable research efforts due to its practical importance on the one hand and its hard tractability by classical robustness concepts on the other hand. Many different robustness concepts for timetabling have been suggested in the literature, some of them very recently. In this paper we compare such concepts on real-world instances. We also introduce a new approach that is generically applicable to any robustness problem. Nevertheless it is able to adapt the special characteristics of the respective problem structure and hence generates solutions that fit to the needs of the respective problem.

Branch and bound algorithms for the bus evacuation problem

The bus evacuation problem (BEP) is a vehicle routing problem that arises in emergency planning. It models the evacuation of a region from a set of collection points to a set of capacitated shelters with the help of buses, minimizing the time needed to bring the last person out of the endangered region. In this work, we describe multiple approaches for finding both lower and upper bounds for the BEP, and apply them in a branch and bound framework. Several node pruning techniques and branching rules are discussed. In computational experiments, we show that solution times of our approach are significantly improved compared to a commercial integer programming solver.

Recovery-to-optimality:A new two-stage approach to robustness with an application to aperiodic timetabling

Abstract The goal of robust optimization is to hedge against uncertainties: in most real-world applications, the specific problem instance depends on uncertain data and is hence not known beforehand. In this work we introduce a new two-stage approach called recovery-to-optimality to handle uncertain optimization problems. Motivated by two-stage stochastic programming and in a similar spirit as the well-known approaches of adjustable robustness or recovery robustness, our new concept allows us to adapt a solution when the realized input scenario is revealed. Using a metric in the solution space measuring the recovery costs, we can evaluate the worst-case costs or the average costs of any solution. Our new concept recovery-to-optimality asks for a solution which can be recovered to an optimal solution with low recovery costs. We set up the robust counterpart (RecOpt) for this concept. However, our intention is to provide a practical approach that can easily be used to generate robust solutions for any application. Building on solution algorithms for the deterministic problem, and on algorithms from location theory, we propose a generic procedure which is able to generate solutions with low recovery costs. We point out properties of these solutions and analyze special cases in which the outcome of the procedure coincides with the optimal solutions to (RecOpt). In an experimental study, we apply our approach to linear programs, and to the problem of finding aperiodic train timetables. We compare it to other robustness concepts, and discuss their trade-offs with respect to multiple evaluation criteria.

A robust bus evacuation model with delayed scenario information

Due to natural or man-made disasters, the evacuation of a whole region or city may become necessary. Apart from private traffic, the emergency services also need to consider transit-dependent evacuees which have to be transported from collection points to secure shelters outside the endangered region with the help of a bus fleet. We consider a simplified version of the arising bus evacuation problem (BEP), which is a vehicle scheduling problem that aims at minimizing the network clearance time, i.e., the time needed until the last person is brought to safety. In this paper, we consider an adjustable robust formulation without recourse for the BEP, the robust bus evacuation problem (RBEP), in which the exact numbers of evacuees are not known in advance. Instead, a set of likely scenarios is known. After some reckoning time, this uncertainty is eliminated and planners are given exact figures. The problem is to decide for each bus, if it is better to send it right away—using uncertain information on the evacuees—or to wait until the the scenario becomes known. We present a mixed-integer linear programming formulation for the RBEP and discuss solution approaches; in particular, we present a tabu search framework for finding heuristic solutions of acceptable quality within short computation time. In computational experiments using both randomly generated instances and the real-world scenario of evacuating the city of Kaiserslautern, Germany, we compare our solution approaches.

Evaluating line concepts using travel times and robustness

Line planning is an early step in the planning process in public transportation, usually followed by designing the timetable. The problems related to both steps are known to be NP-hard, and an integrated model finding a line plan and a timetable simultaneously seems out of scope from a computational point of view. However, the line plan influences also the quality of the timetable to be computed in the next planning step.In this paper we analyze the impact of different line planning models by comparing not only typical characteristics of the line plans, but also their impact on timetables and their robustness against delays. To this end, we set up a simulation platform LinTim which enables us to compute a timetable for each line concept and to experimentally evaluate its performance under delays. Using the German railway intercity network, we evaluate the quality of different line plans from a line planning, a timetabling, and a delay management perspective.

The price of robustness in timetable information

In timetable information in public transport the goal is to search for a good passengers path between an origin and a destination. Usually, the travel time and the number of transfers shall be minimized. In this paper, we consider robust timetable information, i.e. we want to identify a path which will bring the passenger to the planned destination even in the case of delays. The classic notion of strict robustness leads to the problem of identifying those changing activities which will never break in any of the expected delay scenarios. We show that this is in general a strongly NP-hard problem. Therefore, we propose a conservative heuristic which identifies a large subset of these robust changing activities in polynomial time by dynamic programming and so allows us to find strictly robust paths efficiently. We also transfer the notion of light robustness, originally introduced for timetabling, to timetable information. In computational experiments we then study the price of strict and light robustness: How much longer is the travel time of a robust path than of a shortest one according to the published schedule? Based on the schedule of high-speed trains within Germany of 2011, we quantitatively explore the trade-off between the level of guaranteed robustness and the increase in travel time. Strict robustness turns out to be too conservative, while light robustness is promising: a modest level of guarantees is achievable at a reasonable price for the majority of passengers.

Robust load planning of trains in intermodal transportation

In this paper, the problem of robust load planning for trains in intermodal container terminals is studied. The goal of load planning is to choose wagon settings and assign load units to wagons of a train such that the utilization of the train is maximized, and setup and transportation costs in the terminal are minimized. However, in real-world applications, many of the parameters needed for the model are not known exactly. Since feasibility of the resulting load distribution has always to be guaranteed, we decided to use a robust approach. In particular, we apply the concepts of strict and adjustable robustness to enhance the load planning problem. Based on a formulation developed in Bruns and Knust (OR Spectrum 34:511–533, 2012) for the deterministic load planning problem, we propose mixed-integer linear programming formulations for most of the respective robust counterparts, dependent on the type of uncertainty. An experimental study shows that most of the robust problems can be solved within runtimes of a few minutes, which is good enough for real-world applications. Furthermore, our results indicate that robust solutions may improve the planning considerably, and that it is promising to add robustness even to large mixed-integer programs with many and diverse technical constraints.

An experimental comparison of periodic timetabling models

Abstract In the Periodic Timetabling Problem, vehicle arrivals and departures need to be scheduled over a periodically repeating time horizon. Its relevance and applicability have been demonstrated by several real-world implementations, including the Netherlands railways and the Berlin subway. In this work, we consider the practical impact of two possible problem variations: firstly, how passenger paths are handled, and secondly, how line frequencies are included. In computational experiments on real-world and close-to real-world networks, we can show that passenger travel times can significantly benefit from extended models.

A scenario-based approach for robust linear optimization

Finding robust solutions of an optimization problem is an important issue in practice. The established concept of Ben-Tal et al. [2] requires that a robust solution is feasible for all possible scenarios. However, this concept is very conservative and hence may lead to solutions with a bad objective value and is in many cases hard to solve. Thus it is not suitable for most practical applications. In this paper we suggest an algorithm for calculating robust solutions that is easy to implement and not as conservative as the strict robustness approach. We show some theoretical properties of our approach and evaluate it using linear programming problems from NetLib.