Christina Büsing

Recoverable robust knapsacks: the discrete scenario case

The knapsack problem is one of the basic problems in combinatorial optimization. In real-world applications it is often part of a more complex problem. Examples are machine capacities in production planning or bandwidth restrictions in telecommunication network design. Due to unpredictable future settings or erroneous data, parameters of such a subproblem are subject to uncertainties. In high risk situations a robust approach should be chosen to deal with these uncertainties. Unfortunately, classical robust optimization outputs solutions with little profit by prohibiting any adaption of the solution when the actual realization of the uncertain parameters is known. This ignores the fact that in most settings minor changes to a previously determined solution are possible. To overcome these drawbacks we allow a limited recovery of a previously fixed item set as soon as the data are known by deleting at most k items and adding up to ℓ new items. We consider the complexity status of this recoverable robust knapsack problem and extend the classical concept of cover inequalities to obtain stronger polyhedral descriptions. Finally, we present two extensive computational studies to investigate the influence of parameters k and ℓ to the objective and evaluate the effectiveness of our new class of valid inequalities.

Recoverable Robust Shortest Path Problems

In this paper we investigate two different recoverable robust models to deal with cost uncertainties in a shortest path problem. Recoverable robustness extends the classical concept of robustness to deal with uncertainties by incorporating limited recovery actions after the full data are revealed. Our first model focuses on the case where the recovery actions are quite restricted: after a simple path is fixed in the first stage, in the second stage, after all data are revealed, any path containing at most k new arcs may be chosen. Thus, the parameter k can be interpreted as a mediator between robust optimization - no changes allowed - and optimization on the fly - an arbitrary solution can be chosen. Considering three classical scenario sets, which model uncertainties in the cost function, we show that this new problem is strongly NP-hard in all these cases and is not approximable, unless P=NP. This is in contrast to the robust shortest path problem, where, for example, an optimal solution can be computed efficiently for interval and Gamma-scenarios. For series-parallel graphs and interval scenarios, we present a polynomial time algorithm for this recoverable robust setting. In our second model the recovery set, i.e., the set of paths selectable in the second stage is not limited, but deviating from the previous choice comes at extra cost. Thus, a path chosen in the first stage produces renting costs modeled as an alpha-fraction of the scenario cost. For an arc taken in the second stage the remaining cost needs to be paid in addition to some extra inflation cost modeled by a beta-fraction of the scenario cost, if the arc was not reserved beforehand. The complexity status of this problem is similar to the robust case. Yet, for Gamma-scenarios the problem is again strongly NP-hard, but can be approximated.

Recoverable robust knapsacks: Γ-scenarios

In this paper, we investigate the recoverable robust knapsack problem, where the uncertainty of the item weights follows the approach of Bertsimas and Sim [3,4]. In contrast to the robust approach, a limited recovery action is allowed, i.e., up to k items may be removed when the actual weights are known. This problem is motivated by the assignment of traffic nodes to antennas in wireless network planning. Starting from an exponential min-max optimization model, we derive an integer linear programming formulation of quadratic size. In a preliminary computational study, we evaluate the gain of recovery using realistic planning data.

The Exact Subgraph Recoverable Robust Shortest Path Problem

Passengers of a public transportation system are often forced to change their planned route due to deviation in travel times. Rerouting is mostly done by simple means such as announcements. We introduce a model, in which the passenger computes his optimal route on his mobile device in a given subnetwork according to the actual travel times. Those travel times are sent to him as soon as a delay occurs. The main focus of this paper is on the calculation of a small subnetwork. This subnetwork shall contain for every realization of travel times a shortest path of the original network and minimize the number of arcs. For this so called

Robust algorithms for sorting railway cars

\textsc{Exact Subgraph Recoverable Robust Shortest Path}

A New Theoretical Framework for Robust Optimization Under Multi-Band Uncertainty

problem we introduce an approximation algorithm with an approximation factor of

0–1 Multiband Robust Optimization

\frac{m}{\ell}

The budgeted minimum cost flow problem with unit upgrading cost

, for any fixed constant ? ? ?. This is the best possible approximation factor for the interval- and the Γ-scenario case, in which all realizations of travel times are given indirectly by lower and upper bounds on the arc cost. Unless P = NP, for those two scenario sets the problems is not approximable with a factor better than m (1 ? ?), where m is the number of arcs in the given graph and ?> 0.

Line planning, path constrained network flow and inapproximability

We consider a sorting problem from railway optimization called train classification: incoming trains are split up into their single cars and reassembled to form new outgoing trains. Trains are subject to delay, which may turn a prepared sorting schedule infeasible for the disturbed situation. The classification methods applied today deal with this issue by completely disregarding the input order of cars, which provides robustness against any amount of disturbance but also wastes the potential contained in the a priori knowledge about the input. We introduce a new method that provides a feasible sorting schedule for the expected input and allows to flexibly insert additional sorting steps if the schedule has become infeasible after revealing the disturbed input. By excluding disruptions that almost never occur from our consideration, we obtain a classification process that is quicker than the current railway practice but still provides robustness against realistic delays. In fact, our algorithm allows flexibly trading off fast classification against high degrees of robustness depending on the respective need. We further explore this flexibility in experiments on real-world traffic data, underlining our algorithm improves on the methods currently applied in practice.

Robust and sustainable supply chains under market uncertainties and different risk attitudes – A case study of the German biodiesel market

We provide an overview of our main results about studying Linear Programming Problems whose coefficient matrix is subject to uncertainty and the uncertainty is modeled through a multi-band set. Such an uncertainty set generalizes the classical one proposed by Bertsimas and Sim [3] and is particularly suitable in the common case of arbitrary non-symmetric distributions of the parameters. Our investigations were inspired by practical needs of our industrial partner in ongoing projects with focus on the design of robust telecommunications networks.