High Multiplicity Scheduling with Switching Costs for few Products

We study a variant of the single machine capacitated lot-sizing problem with sequence-dependent setup costs and product-dependent inventory costs. We are given a single machine and a set of products associated with a constant demand rate, maximum loading rate and holding costs per time unit. Switching production from one product to another incurs sequencing costs based on the two products. In this work, we show that by considering the high multiplicity setting and switching costs, even trivial cases of the corresponding "normal" counterparts become non-trivial in terms of size and complexity. We present solutions for one and two products.Comment: 10 pages (4 appendix), to be published in Operations Research Proceedings 201

Cyclic Lot-Sizing Problems with Sequencing Costs

We study a single machine lot-sizing problem, where n types of products need to be scheduled on the machine. Each product is associated with a constant demand rate, maximum production rate and inventory costs per time unit. Every time when the machine switches production between products, sequencing costs are incurred. These sequencing costs depend both on the product the machine just produced and the product the machine is about to produce. The goal is to find a cyclic schedule minimizing total average costs, subject to the condition that all demands are satisfied. We establish the complexity of the problem and we prove a number of structural properties largely characterizing optimal solutions. Moreover, we present two algorithms approximating the optimal schedules by augmenting the problem input. Due to the high multiplicity setting, even trivial cases of the corresponding conventional counterparts become highly non-trivial with respect to the output sizes and computational complexity, even without sequencing costs. In particular, the length of an optimal solution can be exponential in the input size of the problem. Nevertheless, our approximation algorithms produce schedules of a polynomial length and with a good quality compared to the optimal schedules of exponential length

Characterizing width two for variants of treewidth

In this paper, we consider the notion of special treewidth, recently introduced by Courcelle (2012). In a special tree decomposition, for each vertex v in a given graph, the bags containing v form a rooted path. We show that the class of graphs of special treewidth at most two is closed under taking minors, and give the complete list of the six minor obstructions. As an intermediate result, we prove that every connected graph of special treewidth at most two can be constructed by arranging blocks of special treewidth at most two in a specific tree-like fashion. Inspired by the notion of special treewidth, we introduce three natural variants of treewidth, namely spaghetti treewidth, strongly chordal treewidth and directed spaghetti treewidth. All these parameters lie between pathwidth and treewidth, and we provide common structural properties on these parameters. For each parameter, we prove that the class of graphs having the parameter at most two is minor closed, and we characterize those classes in terms of a tree of cycles with additional conditions. Finally, we show that for each k≥3, the class of graphs with special treewidth, spaghetti treewidth, directed spaghetti treewidth, or strongly chordal treewidth, respectively at most k, is not closed under taking minors