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We consider the problem to evacuate several regions due to river flooding, where sufficient time is given to plan ahead. To ensure a smooth evacuation procedure, our model includes the decision which regions to assign to which shelter, and when evacuation orders should be issued, such that roads do not become congested.
Due to uncertainty in weather forecast, several possible scenarios are simultaneously considered in a robust optimization framework. To solve the resulting integer program, we apply a Tabu search algorithm based on decomposing the problem into better tractable subproblems. Computational experiments on random instances and an instance based on Kulmbach, Germany, data show considerable improvement compared to an MIP solver provided with a strong starting solution.

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value.
A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately.
While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.
Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time.

The ordered weighted averaging objective (OWA) is an aggregate function over multiple optimization criteria which received increasing attention by the research community over the last decade. Different to the ordered weighted sum, weights are attached to ordered objective functions (i.e., a weight for the largest value, a weight for the second-largest value and so on). As this contains max-min or worst-case optimization as a special case, OWA can also be considered as an alternative approach to robust optimization.
For linear programs with OWA objective, compact reformulations exist, which result in extended linear programs. We present new such reformulation models with reduced size. A computational comparison indicates that these formulations improve solution times.

We discuss the problem of evaluating a robust solution.
To this end, we first give a short primer on how to apply robustification approaches to uncertain optimization problems using the assignment problem and the knapsack problem as illustrative examples.
As it is not immediately clear in practice which such robustness approach is suitable for the problem at hand,
we present current approaches for evaluating and comparing robustness from the literature, and introduce the new concept of a scenario curve. Using the methods presented in this paper, an easy guide is given to the decision maker to find, solve and compare the best robust optimization method for his purposes.

Geometric Programming is a useful tool with a wide range of applications in engineering. As in real-world problems input data is likely to be affected by uncertainty, Hsiung, Kim, and Boyd introduced robust geometric programming to include the uncertainty in the optimization process. They also developed a tractable approximation method to tackle this problem. Further, they pose the question whether there exists a tractable reformulation of their robust geometric programming model instead of only an approximation method. We give a negative answer to this question by showing that robust geometric programming is co-NP hard in its natural posynomial form.

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad on average. A solution that optimizes the average performance on the other hand lacks in worst-case performance guarantee.
In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance.
Building upon a theoretical observation on the structure of Pareto solutions for problems with polyhedral feasible sets, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectivity of both the proposed algorithm, and the bicriteria perspective in general.

Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-hard minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist.
A very easy but effective approximation technique is to compute the midpoint solution of the original optimization problem, which aims at optimizing the average regret, and also the average nominal objective. It is a well-known result that the regret of the midpoint solution is at most 2 times the optimal regret. Besides some academic instances showing that this bound is tight, most instances reveal a way better approximation ratio.
We introduce a new lower bound for the optimal value of the minmax regret problem. Using this lower bound we state an algorithm that gives an instance dependent performance guarantee of the midpoint solution for combinatorial problems that is at most 2. The computational complexity of the algorithm depends on the minmax regret problem under consideration; we show that the sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems.
To illustrate the quality of the proposed bound, we use it within a branch and bound framework for the robust shortest path problem. In an experimental study comparing this approach with a bound from the literature, we find a considerable improvement in computation times.

We consider a network flow problem, where the outgoing flow is reduced by a certain percentage in each node. Given a maximum amount of flow that can leave the source node, the aim is to find a solution that maximizes the amount of flow which arrives at the sink.
Starting from this basic model, we include two new, additional aspects: On the one hand, we are able to reduce the loss at some of the nodes; on the other hand, the exact loss values are not known, but may come from a discrete uncertainty set of exponential size.
Applications for problems of this type can be found in evacuation planning, where one would like to improve the safety of nodes such that the number of evacuees reaching safety is maximized.
We formulate the resulting robust flow problem with losses and improvability as a mixed-integer program for finitely many scenarios, and present an iterative scenario-generation procedure that avoids the inclusion of all scenarios from the beginning. In a computational study using both randomly generated instance and realistic data based on the city of Nice, France, we compare our solution algorithms.

We consider the problem of evacuating an urban area caused by a natural or man-made disaster. There are several planning aspects that need to be considered in such a scenario, which are usually considered separately, due to their computational complexity. These aspects include: Which shelters are used to accommodate evacuees? How to schedule public transport for transit-dependent evacuees? And how do public and individual traffic interact? Furthermore, besides evacuation time, also the risk of the evacuation needs to be considered.
We propose a macroscopic multi-criteria optimization model that includes all of these questions simultaneously. As a mixed-integer programming formulation cannot handle instances of real-world size, we develop a genetic algorithm of NSGA-II type that is able to generate feasible solutions of good quality in reasonable computation times.
We extend the applicability of these methods by also considering how to aggregate instance data, and how to generate solutions for the original instance starting from a reduced solution.
In computational experiments using real-world data modelling the cities of Nice in France and Kaiserslautern in Germany, we demonstrate the effectiveness of our approach and compare the trade-off between different levels of data aggregation.

We consider the problem of scheduling a bus fleet to evacuate persons from an endangered region. As most of the planning data is subject to uncertainty, we develop a two-stage bicriteria robust formulation, which considers both the evacuation time, and the vulnerability of the schedule to changing evacuation circumstances.
As the resulting integer program is too large to solve it directly using an off-the-shelf solver, we develop an iterative algorithm that successively adds new scenarios to the currently considered subproblem. In computational experiments, we show that this approach is fast enough to deal with an instance modeling an evacuation case within the city of Kaiserslautern, Germany.

By natural or man-made disasters, the evacuation of a whole region or city may become necessary. Apart from private traffic, the evacuation from collection points to secure shelters outside the endangered region will be realized by a bus fleet made available by emergency relief. The arising Bus Evacuation Problem (BEP) is a vehicle scheduling problem, in which a given number of evacuees needs to be transported from a set of collection points to a set of capacitated shelters, minimizing the total evacuation time, i.e., the time needed until the last person is brought to safety.
In this paper we consider an extended version of the BEP, the Robust Bus Evacuation Problem (RBEP), in which the exact numbers of evacuees are not known, but may stem from a set of probable scenarios. However, after a given 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 numbers of evacuees -- or to wait until the numbers become 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, we compare our solution approaches.

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.

We consider the problem of evacuating a region with the help of buses. For a given set of possible collection points where evacuees gather, and possible shelter locations where evacuees are brought to, we need to determine both collection points and shelters we would like to use, and bus routes that evacuate the region in minimum time.
We model this integrated problem using an integer linear program, and present a branch-cut-and-price algorithm that generates bus tours in its pricing step. In computational experiments we show that our approach is able to solve instances of realistic size in sufficient time for practical application, and considerably outperforms the usage of a generic ILP solver.

We argue that the concepts of resilience in engineering science and robustness in mathematical optimization are strongly related. Using evacuation planning as an example application, we demonstrate optimization techniques to improve solution resilience. These include a direct modelling of the uncertainty for stochastic or robust optimization, as well as taking multiple objective functions into account.

We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker.
We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding.

We consider storage loading problems where items with uncertain weights have
to be loaded into a storage area, taking into account stacking and
payload constraints. Following the robust optimization paradigm, we propose
strict and adjustable optimization models for finite and interval-based
uncertainties. To solve these problems, exact decomposition and heuristic
solution algorithms are developed.
For strict robustness, we also present a compact formulation based
on a characterization of worst-case scenarios.
Computational results show that computation times and algorithm
gaps are reasonable for practical applications.
Furthermore, we find that the robustness concepts show different
potential depending on the type of data being used.