Stan P. M. van Hoesel

Economic lot sizing: an O ( n log n ) algorithm that runs in linear time in the Wagner-Whitin case

We consider the n-period economic lot sizing problem, where the cost coefficients are not restricted in sign. In their seminal paper, H. M. Wagner and T. M. Whitin proposed an O(n2) algorithm for the special case of this problem, where the marginal production costs are equal in all periods and the unit holding costs are nonnegative. It is well known that their approach can also be used to solve the general problem, without affecting the complexity of the algorithm. In this paper, we present an algorithm to solve the economic lot sizing problem in O(n log n) time, and we show how the Wagner-Whitin case can even be solved in linear time. Our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure. Furthermore, we show how Wagner and Whitins and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem.

Models and solution techniques for frequency assignment problems

Abstract Wireless communication is used in many different situations such as mobile telephony, radio and TV broadcasting, satellite communication, wireless LANs, and military operations. In each of these situations a frequency assignment problem arises with application specific characteristics. Researchers have developed different modeling ideas for each of the features of the problem, such as the handling of interference among radio signals, the availability of frequencies, and the optimization criterion. This survey gives an overview of the models and methods that the literature provides on the topic. We present a broad description of the practical settings in which frequency assignment is applied. We also present a classification of the different models and formulations described in the literature, such that the common features of the models are emphasized. The solution methods are divided in two parts. Optimization and lower bounding techniques on the one hand, and heuristic search techniques on the other hand. The literature is classified according to the used methods. Again, we emphasize the common features, used in the different papers. The quality of the solution methods is compared, whenever possible, on publicly available benchmark instances.

Routing Trains through railway stations: model formulation and algorithms

In this paper we consider the problem of routing trains through railway stations. This problem occurs as a subproblem in a project which the authors are carrying out in cooperation with the Dutch railways. The project involves the analysis of future infrastructural capacity requirements in the Dutch railway network. Part of this project is the automatic generation and evaluation of timetables. To generate a timetable a hierarchical approach is followed: at the upper level in the hierarchy a tentative timetable is generated, taking into account the specific scheduling problems of the trains at the railway stations at an aggregate level. At the lower level in the hierarchy it is checked whether the tentative timetable is feasible with respect to the safety rules and the connection requirements at the stations. To carry out this consistency check, detailed schedules for the trains at the railway yards have to be generated. In this paper we present a mathematical model formulation for this detailed scheduling problem, based on the Node Packing Problem (NPP). Furthermore, we describe a solution procedure for the problem, based on a branch-and-cut approach. The approach is tested in an empirical study with data from the station of Zwolle in The Netherlands.

Routing trains through a railway station based on a node packing model

In this paper we describe the problem of routing trains through a railway station. This routing problem is a subproblem of the automatic generation of timetables for the Dutch railway system. The problem of routing trains through a railway station is the problem of assigning each of the involved trains to a route through the railway station, given the detailed layout of the railway network within the station and given the arrival and departure times of the trains. When solving this routing problem, several aspects such as capacity, safety, and customer service have to be taken into account. In this paper we describe this routing problem in terms of a Weighted Node Packing Problem. Furthermore, we describe an algorithm for solving this routing problem to optimality. The algorithm is based on preprocessing, valid inequalities, and a branch-and-cut approach. The preprocessing techniques aim at identifying super uous nodes which can be removed from the problem instance. The characteristics of the preprocessing techniques with respect to propagation are investigated. We also present the results of a computational study in which the model, the preprocessing techniques and the algorithm are tested based on data related to the railway stations Arnhem, Hoorn and Utrecht in the Netherlands.

The partial constraint satisfaction problem: Facets and lifting theorems

We study the polyhedral structure of the partial constraint satisfaction problem (PCSP). Among the problems that can be formulated as such are the maximum satisfiability problem and a fairly general model of frequency assignment problems. We present lifting theorems and classes of facet defining inequalities, and we provide preliminary experiments.

A Branch-and-Cut Approach for Solving Railway Line-Planning Problems

An important strategic phase in the planning process of a railway operator is the development of a line plan, i.e., a set of routes (paths) in a network of tracks, operated at a given hourly frequency. We consider a model formulation of the line-planning problem where total operating costs are to be minimized. This model is solved with a branch-and-cut approach, for which we develop a variety of valid inequalities and reduction methods. A computational study of five real-life instances based on examples from Netherlands Railways (NS) is included.

Integrated Lot Sizing in Serial Supply Chains with Production Capacities

We consider a model for a serial supply chain in which production, inventory, and transportation decisions are integrated in the presence of production capacities and concave cost functions. The model we study generalizes the uncapacitated serial single-item multilevel economic lot-sizing model by adding stationary production capacities at the manufacturer level. We present algorithms with a running time that is polynomial in the planning horizon when all cost functions are concave. In addition, we consider different transportation and inventory holding cost structures that yield improved running times: inventory holding cost functions that are linear and transportation cost functions that are either linear, or are concave with a fixed-charge structure. In the latter case, we make the additional common and reasonable assumption that the variable transportation and inventory costs are such that holding inventories at higher levels in the supply chain is more attractive from a variable cost perspective. While the running times of the algorithms are exponential in the number of levels in the supply chain in the general concave cost case, the running times are remarkably insensitive to the number of levels for the other two cost structures.

On the complexity of postoptimality analysis of 0/1 programs

Abstract In this paper we address the complexity of postoptimality analysis of 0 1 programs with a linear objective function. After an optimal solution has been determined for a given cost vector, one may want to know how much each cost coefficient can vary individually without affecting the optimality of the solution. We show that, under mild conditions, the existence of a polynomial method to calculate these maximal ranges implies a polynomial method to solve the 0 1 program itself. As a consequence, postoptimality analysis of many well-known NP-hard problems cannot be performed by polynomial methods, unless P =NP. A natural question that arises with respect to these problems is whether it is possible to calculate in polynomial time reasonable approximations of the maximal ranges. We show that it is equally unlikely that there exists a polynomial method that calculates conservative ranges for which the relative deviation from the true ranges is guaranteed to be at most some constant. Finally, we address the issue of postoptimality analysis ofe-optimal solutions of NP-hard 0 1 problems. It is shown that for an e-optimal solution that has been determined in polynomial time, it is not possible to calculate in polynomial time the maximal amount by which a cost coefficient can be increased such that the solution remains e-optimal, unless P = NP.

On solving multi-type railway line planning problems

An important strategic element in the planning process of a railway operator is the development of a line plan, i.e., a set of routes (paths) on the network of tracks, operated at a given hourly frequency. The models described in the literature have thus far considered only lines that halt at all stations along their route. In this paper we introduce several models for solving line planning problems in which lines can have different halting patterns. Correctness and equivalence proofs for these models are given, as well as an evaluation using several real-life instances.

Solving partial constraint satisfaction problems with tree decomposition

In this paper, we describe a computational study to solve hard partial constraint satisfaction problems (PCSPs) to optimality. The PCSP is a general class of problems that contains a diversity of problems, such as generalized subgraph problems, MAX-SAT, Boolean quadratic programs, and assignment problems like coloring and frequency planning. We present a dynamic programming algorithm that solves PCSPs based on the structure (tree decomposition) of the underlying constraint graph. With the use of dominance and bounding techniques, we are able to solve small and medium-size instances of the problem to optimality and to obtain good lower bounds for large-size instances within reasonable time and memory limits.