Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Vehicle routing on real road networks

The vehicle routing problem (VRP) has received particular attention, in the field of transportation and logistics. Producing good solutions for the problem is of interest both commercially and theoretically. Reliable solutions to real life applications require an approach based on realistic assumptions that resemble real-world conditions. In that respects, this thesis studies vehicle routing problems on real road networks addressing aspects of the problem that need to be modelled on the original road network graph and aims to provide appropriate modelling techniques for solving them. As a preliminary step, chapter 2 studies the travelling salesman problem (TSP) on real road networks, referred to as the Steiner TSP (STSP) and proposes alternative integer programming formulations for the problem and some other related routing problems. The performances of formulations is examined both theoretically and computationally. Chapter 3 highlights the fact that travel speeds on road networks are correlated and uses a real traffic dataset to explore the structure of this correlation. In conclusion, it is shown that there is still significant spatial correlations between speeds on roads that are up to twenty links apart, in our congested road network. Chapter 4 extends chapter 2 and incorporates the findings of chapter 3 into a modelling framework for VRP. The STSP with correlated costs is defined as a potentially useful variant of VRP that considers the costs in the STSP to be stochastic random variables with correlation. The problem is then formulated as a single-objective problem with eight different integer programming formulations presented. It is then shown how to account for three different correlation structures in each of the formulations. Chapter 5 considers the VRPs with time windows and shows how most of the exact algorithms proposed for them, might not be applicable if the problem is defined on the original road network graph due to the underlying assumption of these algorithms that the cheapest path between a pair of customers is the same as the quickest path. This assumption is not always true on a real road network. Instead some alternative pricing routines are proposed that can solve the problem directly on the original graph

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations