Anita Schöbel

Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.

A Model for the Delay Management Problem based on Mixed-Integer-Programming

Abstract Dealing with delayed vehicles is a necessary issue in the dispositive work of a public transportation company. If a vehicle arrives at some station with a delay, it has to be decided if the connecting vehicles should wait for changing passengers or if they should depart in time. A possible objective function is to minimize the sum of all delays over all customers using the transportation network. In this paper the delay management problem is formulated as a mixed integer linear program, and solution approaches based on this formulation are indicated.

Integer programming approaches for solving the delay management problem

The delay management problem deals with reactions in case of delays in public transportation. More specifically, the aim is to decide if connecting vehicles should wait for delayed feeder vehicles or if it is better to depart on time. As objective we consider the convenience over all customers, expressed as the average delay of a customer when arriving at his or her destination. We present path-based and activity-based integer programming models for the delay management problem and show the equivalence of these formulations. Based on these, we present a simplification of the (cubic) activity-based model which results in an integer linear program. We identify cases in which this linearization is correct, namely if the so-called never-meet property holds. We analyze this property using real-world railway data. Finally, we show how to find an optimal solution in linear time if the never-meet property holds.

To Wait or Not to Wait---And Who Goes First? Delay Management with Priority Decisions

Delay management is an important issue in the daily operations of any railway company. The task is to update the planned timetable to a disposition timetable in such a way that the inconvenience for the passengers is as small as possible. The two main decisions that have to be made in this respect are the wait-depart decisions, to decide which connections should be maintained in case of delays, and the priority decisions, which determine the order in which trains are allowed to pass a specific piece of track. The latter are necessary to take the limited capacity of the track system into account. While the wait-depart decisions have been intensively studied in the literature, the priority decisions in the capacitated case have been neglected so far in delay management optimization models. In the current paper, we add the priority decisions to the integer programming formulation of the delay management problem and are hence able to deal with the capacitated case. The corresponding constraints are disjunctive constraints leading to cycles in the resulting event-activity network. Nevertheless, we are able to derive reduction techniques for the network that enable us to extend the formulation of the never-meet property from the uncapacitated delay management problem to the capacitated case. We then use our results to derive exact and heuristic solution procedures for solving the delay management problem. The results of the algorithms are evaluated both from a theoretical and a numerical point of view. The latter has been done within a case study using the railway network in the region of Harz, Germany.

Line planning in public transportation: models and methods

The problem of defining suitable lines in a public transportation system (bus, railway, tram, or underground) is an important real-world problem that has also been well researched in theory. Driven by applications, it often lacks a clear description, but is rather stated in an informal way. This leads to a variety of different published line planning models. In this paper, we introduce some of the basic line planning models, identify their characteristics, and review literature on models, mathematical approaches, and algorithms for line planning. Moreover, we point out related topics as well as current and future directions of research.

Hub Location Problems in Urban Traffic Networks

In this paper we present new hub location models which are applicable for urban public transportation networks. In order to obtain such models we relax some of the general assumptions that are usually satisfied in hub location problems, but which are not useful for public transportation networks. For instance we do not require that the hub nodes have to be completely interconnected. These new models are based on network design formulations, in which the constraint that all flow has to be routed via some hub nodes is formulated by a flow conservation law. We present some solution approaches for these new models and illustrate the results on a numerical example.

Capacity constraints in delay management

We consider (small) disturbances of a railway system. In case of such delays, one has to decide if connecting trains should wait for delayed feeder trains or if they should depart on time, i.e. which connections should be maintained and which can be dropped. Finding such wait-depart decisions (minimizing e.g. the average delay of the passengers) is called the delay management problem. In the literature, the limited capacity of the tracks (meaning that no two trains can use the same piece of track at the same time) has so far been neglected in the delay management problem. In this paper we present models and first results integrating these important constraints. We develop algorithmic approaches that have been tested at a real-world example provided by Deutsche Bahn AG.

Recoverable Robustness in Shunting and Timetabling

In practical optimization problems, disturbances to a given instance are unavoidable due to unpredictable events which can occur when the system is running. In order to face these situations, many approaches have been proposed during the last years in the area of robust optimization. The basic idea of robustness is to provide a solution which is able to keep feasibility even if the input instance is disturbed, at the cost of optimality. However, the notion of robustness in every day life is much broader than that pursued in the area of robust optimization so far. In fact, robustness is not always suitable unless some recovery strategies are introduced. Recovery strategies are some capabilities that can be used when disturbing events occur, in order to keep the feasibility of the pre-computed solution. This suggests to study robustness and recoverability in a unified framework. Recently, a first tentative of unifying the notions of robustness and recoverability into a new integrated notion of recoverable robustness has been done in the context of railway optimization. In this paper, we review the recent algorithmic results achieved within the recoverable robustness model in order to evaluate the effectiveness of this model. To this aim, we concentrate our attention on two problems arising in the area of railway optimization: the shunting problem and the timetabling problem. The former problem regards the reordering of freight train cars over hump yards while the latter one consists in finding passenger train timetables in order to minimize the overall passengers traveling time. We also report on a generalization of recoverable robustness called multi-stage recoverable robustness which aims to extend recoverable robustness when multiple recovery phases are required.

Continuous location of dimensional structures

Abstract A natural extension of point facility location problems are those problems in which facilities are extensive, i.e. those that cannot be represented by isolated points but as some dimensional structures, such as straight lines, line-segments, polygonal curves, or circles. In this paper a review of the existing work on the location of extensive facilities in continuous spaces is given. Gaps in the knowledge are identified and suggestions for further research are made.

To Wait or Not to Wait? The Bicriteria Delay Management Problem in Public Transportation

Assume that a train reaches a station with delay. At the station there is a bus ready to depart. The question of whether the bus should wait for the delayed train or depart on time is called the delay management problem. Different single objective functions for this problem have been introduced and analyzed. In this paper, we present a bicriteria model for the delay management problem, taking into account both the delay of the vehicles and the number of passengers who miss a connection. Our model does not depend on detailed data about the passengers and hence can be easily implemented in practice. To analyze the problem, we present an integer programming formulation and a graph-theoretic approach that is based on discrete time/cost trade-off project networks. Using results of project planning, we develop an efficient solution method. We tested our procedure using real-world data. The results show the applicability of the approach.