An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

Ising formulations of many NP problems

We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more references; v3: substantial revision and extension, to-be-published versio

Reformulated acyclic partitioning for rail-rail containers transshipment

Many rail terminals have loading areas that are properly equipped to move containers between trains. With the growing throughput of these terminals all the trains involved in a sequence of such movements may not ¿t in the loading area simultaneously, and storage areas are needed to place containers waiting for their destination train, although this storage increases the cost of the transshipment. This increases the complexity of the planning decisions concerning these activities, since now trains need to be packed in groups that ¿t in the loading area, in such a way that the number of containers moved to the storage area is minimized. Additionally, each train is only allowed to enter the loading area once. Similarly to previous authors, we model this situation as an acyclic graph partitioning problem for which we present a new formulation, and several valid inequalities based on its theoretical properties. Our computational experiments show that the new formulation outperforms the previously existing ones, providing results that improve even on the best exact algorithm designed so far for this problem.Peer ReviewedPostprint (author's final draft

Two novel evolutionary formulations of the graph coloring problem

We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The first formulation is based on the relationship that exists between a graph's chromatic number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The second formulation, unlike the first one, does not tackle one graph at a time, but rather aims at evolving a `program' to color all graphs belonging to a class whose members all have the same number of nodes and other common attributes. The heuristics that result from these formulations have been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and have been found to be competitive when compared to the several other heuristics that have also been tested on those graphs.Comment: To appear in Journal of Combinatorial Optimizatio