Caj Cor Hurkens

Time-Indexed Formulations for Machine Scheduling Problems: Column Generation

Time-indexed formulations for machine scheduling problems have received a great deal of attention; not only do the linear programming relaxations provide strong lower bounds, but they are good guides for approximation algorithms as well. Unfortunately, time-indexed formulations have one major disadvantage--their size. Even for relatively small instances the number of constraints and the number of variables can be large. In this paper, we discuss how Dantzig-Wolfe decomposition techniques can be applied to alleviate, at least partly, the difficulties associated with the size of time-indexed formulations. In addition, we show that the application of these techniques still allows the use of cut generation techniques.

Linear programming models with planned lead times for supply chain operations planning

This paper contributes to the development of models for capacity constrained Supply Chain Operations Planning (SCOP). We focus on production environments with arbitrary supply chain structures. The demand for the end items is assumed to be exogenously determined. We solve the SCOP problem with Linear Programming models using planned lead times with multi-period capacity consumption. Using planned lead times increases the reliability of the communication between SCOP and Scheduling with regard to the feasibility of the planning. Planned lead times also reduce the nervousness in the system. We model capacity constraints on the quantity of items that can be assembled within a time interval. In particular, items can be assigned to multiple resources. We discuss two LP approaches which plan the production of items so that a sum of inventory costs and costs due to backordering is minimized.

Market Split and Basis Reduction: Towards a Solution of the Cornuéjols-Dawande Instances

At the IPCO VI conference CornuA©jols and Dawande proposed a set of 0-1 linear programming instances that proved to be very hard to solve by traditional methods, and in particular by linear programming-based branch-and-bound. They offered thesemarket split instances as a challenge to the integer programming community. The market split problem can be formulated as a system of linear diophantine equations in 0-1 variables. In our study we use the algorithm of Aardal, Hurkens, and Lenstra (1998) based on lattice basis reduction. This algorithm is not restricted to deal with market split instances only but is a general method for solving systems of linear diophantine equations with bounds on the variables. We show computational results from solving both feasibility and optimization versions of the market split instances with up to 7 equations and 60 variables and discuss various branching strategies and their effect on the number of enumerated nodes. To our knowledge, the largest feasibility and optimization instances solved before had 6 equations and 50 variables, and 4 equations and 30 variables, respectively. We also present a probabilistic analysis describing how to compute the probability of generating infeasible market split instances. By generating instances the way prescribed by CornuA©jols and Dawande, one obtains relatively many feasible instances for sizes larger than 5 equations and 40 variables.

An improved MIP-based approach for a multi-skill workforce scheduling problem

This paper deals with scheduling complex tasks with an inhomogeneous set of resources. The problem is to assign technicians to tasks with multi-level skill requirements. Here, the requirements are merely the presence of a set of technicians that possess the necessary capabilities. An additional complication is that a set of combined technicians stays together for the duration of a work day. This typically applies to scheduling of maintenance and installation operations. We build schedules by repeated application of a flexible matching model that selects tasks to be processed and forms groups of technicians assigned to combinations of tasks. The underlying mixed integer programming (MIP) model is capable of revising technician-task allocations and performs very well, especially in the case of rare skills.

Virtual Private Network Design: A Proof of the Tree Routing Conjecture on Ring Networks

A basic question in virtual private network (VPN) design is if the symmetric version of the problem always has an optimal solution which is a tree network. An affirmative answer would imply that the symmetric VPN problem is solvable in polynomial time. We give an affirmative answer in case the communication network, within which the VPN must be created, is a circuit. This seems to be an important step towards answering the general question. The proof relies on a dual pair of linear programs and actually implies an even stronger property of VPNs. We show that this property also holds for some other special cases of the problem, in particular when the network is a tree of rings.

Experimental Comparison of Approximation Algorithms for Scheduling Unrelated Parallel Machines

This paper presents an empirical comparison of polynomial-time approximation algorithms and local search heuristics for the problem of minimizing total weighted completion time on unrelated parallel machines. Algorithms with a worst-case performance guarantee are based on rounding a fractional solution to an LP-relaxation or to a convex quadratic-programming relaxation. We also investigate dominance relations among the lower bounds resulting from these relaxations.

Local search algorithms for the radio link frequency assignment problem

Radio link frequency assignment problems arise in practice when a network of radio links has to be established. Each radio link has to be assigned an operating frequency. The frequency assignments have to comply with certain regulations and physical characteristics of the transmitters. The interference level between frequencies assigned to different links has to be small, since otherwise communication will be distorted. Two objectives are considered: the total interference costs and, in case of no interference, the frequency spectrum usage. We develop several approximation algorithms for the problem and compare their performance on some practical instances. We also develop a lower bounding technique based on a nonlinear relaxation in order to estimate the quality of the approximate solutions for some of these instances.

On the nearest neighbor rule for the traveling salesman problem

Rosenkrantz et al. (SIAM J. Comput. 6 (1977) 563) and Johnson and Papadimitriou (in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, Chichester, 1985, pp. 145-180, (Chapter 5)) constructed families of TSP instances with n cities for which the nearest neighbor rule yields a tour-length that is a factor @W(logn) above the length of the optimal tour. We describe two new families of TSP instances, for which the nearest neighbor rule shows the same bad behavior. The instances in the first family are graphical, and the instances in the second family are Euclidean. Our construction and our arguments are extremely simple and suitable for classroom use.

Timetabling problems at the TU Eindhoven

The students of the Department of Industrial Design at the TU Eindhoven are allowed to design part of their curriculum by selecting courses from a huge course pool. They do this by handing in ordered preference lists with their favorite courses for the forthcoming time period. Based on this information (and on many other constraints), the department then assigns courses to students. Until recently, the assignment was computed by human schedulers who used a quite straightforward greedy approach. In 2005, however, the number of students increased substantially, and as a consequence the greedy approach did not yield acceptable results anymore. This paper discusses the solution of this real-world timetabling problem. We present a complete mathematical formulation and explain all the constraints resulting from the situation in Eindhoven. We solve the problem using lexicographical optimization with four subproblems. For all four subproblems, an elegant integer linear programming model is given which easily can be put into CPLEX. Finally, we report on our computational experiments and results around the Eindhoven real-world data.

Blowing up convex sets in the plane

Abstract Let K be a convex set in R 2 such that every line in R 2 meets K+ Z 2 . We prove that αK+ Z 2 = R 2 for α⩾1+ 2 3 √3 , and that this bound is best possible, thus solving a problem of Kannan and Lovasz.