Felix G. König

Solutions to real-world instances of PSPACE-complete stacking

We investigate a complex stacking problem that stems from storage planning of steel slabs in integrated steel production. Besides the practical importance of such stacking tasks, they are appealing from a theoretical point of view. We show that already a simple version of our stacking problem is PSPACE-complete. Thus, fast algorithms for computing provably good solutions as they are required for practical purposes raise various algorithmic challenges. We describe an algorithm that computes solutions within 5/4 of optimality for all our real-world test instances. The basic idea is a search in an exponential state space that is guided by a state-valuation function. The algorithm is extremely fast and solves practical instances within a few seconds. We assess the quality of our solutions by computing instance-dependent lower bounds from a combinatorial relaxation formulated as mixed integer program. To the best of our knowledge, this is the first approach that provides quality guarantees for such problems.

Approximation Algorithms for Capacitated Location Routing

An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of capacitated location routing, an important generalization of vehicle routing where the cost of opening the depots from which vehicles operate is taken into account. Our results originate from combining algorithms and lower bounds for different relaxations of the original problem; along with location routing we also obtain approximation algorithms for multidepot capacitated vehicle routing by this framework. Moreover, we extend our results to further generalizations of both problems, including a prize-collecting variant, a group version, and a variant where cross-docking is allowed. We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances. Our study reveals that the quality of the computed solutions is much closer to optimality than the provable approximation factor.

Integrated Sequencing and Scheduling in Coil Coating

We consider a complex planning problem in integrated steel production. A sequence of coils of sheet metal needs to be color coated in consecutive stages. Different coil geometries and changes of colors necessitate time-consuming setup work. In most coating stages one can choose between two parallel color tanks. This can either reduce the number of setups needed or enable setups concurrent with production. A production plan comprises the sequencing of coils and the scheduling of color tanks and setup work. The aim is to minimize the makespan for a given set of coils. We present an optimization model for this integrated sequencing and scheduling problem. A core component is a graph theoretical model for concurrent setup scheduling. It is instrumental for building a fast heuristic that is embedded into a genetic algorithm to solve the sequencing problem. The quality of our solutions is evaluated via an integer program based on a combinatorial relaxation, showing that our solutions are within 10% of the optimum. Our algorithm is implemented at Salzgitter Flachstahl GmbH, a major German steel producer. This has led to an average reduction in makespan by over 13% and has greatly exceeded expectations. This paper was accepted by Dimitris Bertsimas, optimization.

Sorting with Complete Networks of Stacks

Knuth introduced the problem of sorting numbers using a sequence of stacks. Tarjan extended this idea to sorting with acyclic networks of stacks (and queues), where items to be sorted move from a source through the network to a sink while they may be stored temporarily at nodes (the stacks). Both characterized which permutations are sortable this way; but they ignored the associated optimization problem (minimize the number of moves) and its complexity. Given a complete, thus cyclic, network of k ≥ 2 stacks, any permutation is obviously sortable. The important question now is how to actually sort with a minimum number of shuffles, i.e., moves in between stacks. This is a natural algorithmic complement to the structural questions asked by Knuth, Tarjan, and others. It is the first time shuffles are considered in stack sorting--despite of the great practical importance of this optimization version. We show that it is NP-hard to approximate the minimum number of shuffles within

Scheduling and packing malleable and parallel tasks with precedence constraints of bounded width

\mathcal{O}(n^{1-\epsilon})

An Integrated Approach to Tactical Transportation Planning in Logistics Networks

for networks of k ≥ 4 stacks, even when the problem is restricted to complete networks, by relating stack sorting to Min k -Partition on circle graphs (for which we prove a stronger inapproximability result of independent interest). For complete networks, a simple merge sort algorithm achieves an approximation ratio of

Scheduling and packing malleable tasks with precedence constraints of bounded width

\mathcal{O}(n \log n)

Multi-Dimensional Commodity Covering for Tariff Selection in Transportation

for k ≥ 3; however, closing the logarithmic gap to our lower bound appears to be an intriguing open question. Yet, on the positive side, we present a tight approximation algorithm which computes a solution with a linear approximation guarantee, using a resource augmentation to αk + 1 stacks, given an α-approximation algorithm for coloring circle graphs. When there are constraints as to which items may be placed on top of each other, deciding about sortability becomes non-trivial again. We show that this problem is PSPACE-complete, for every fixed k ≥ 3.

Sorting with Objectives - Graph-Theoretic Concepts in Industrial Optimization

We study the problems of non-preemptively scheduling and packing malleable and parallel tasks with precedence constraints to minimize the makespan. In the scheduling variant, we allow the free choice of processors; in packing, each task must be assigned to a contiguous subset. Malleable tasks can be processed on different numbers of processors with varying processing times, while parallel tasks require a fixed number of processors.For precedence constraints of bounded width, we resolve the complexity status of the problem with any number of processors and any width bound. We present an FPTAS based on Dilworth’s decomposition theorem for the NP-hard problem variants, and exact efficient algorithms for all remaining special cases. For our positive results, we do not require the otherwise common monotonous penalty assumption on the processing times of malleable tasks, whereas our hardness results hold even when assuming this restriction. We complement our results by showing that these problems are all strongly NP-hard under precedence constraints which form a tree.

Traffic Optimization Under Route Constraints with Lagrangian Relaxation and Cutting Plane Methods

We propose a new mathematical model for transport optimization in logistics networks on the tactical level. The main features include accurately modeled tariff structures and the integration of spatial and temporal consolidation effects via a cyclic pattern expansion. Using several graph-based gadgets, we are able to formulate our problem as a capacitated network design problem. To solve the model, we propose a local search procedure that reroutes flow of multiple commodities at once. Initial solutions are generated by various heuristics, relying on shortest path augmentations and LP techniques. As an important subproblem we identify the optimization of tariff selection on individual links, which we prove to be NP -hard and for which we derive exact as well as fast greedy approaches. We complement our heuristics by lower bounds from an aggregated mixed-integer programming formulation with strengthened inequalities. In a case study from the automotive, chemical, and retail industries, we prove that most of our solutions are within a single-digit percentage of the optimum.