We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchand-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branch-andbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1

We present DRIVE (Dynamic Routing of Independent VEhicles), a planning module to be incorporated in a decision support system for the direct transportation at Van Gend & Loos BV. Van Gend & Loos BV is the largest company providing road transportation in the Benelux with about 1400 vehicles transporting 160,000 packages from thousands of senders to tens of thousands of addressees per day. The heart of DRIVE is a branch-and-price algorithm. Approximation and incomplete optimization techniques as well as a sophisticated column management scheme have been employed to create the right balance between solution speed and solution quality. DRIVE has been tested by simulating a dynamic planning environment with real-life data and has produced very encouraging results.

Dantzig-Wolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

Routing with time windows (VRPTW) has been an area of research that have attracted many researchers within the last 10 { 15 years. In this period a number of papers and technical reports have been published on the exact solution of the VRPTW. The VRPTW is a generalization of the well-known capacitated routing problem (VRP or CVRP). In the VRP a eet of vehicles must visit (service) a number of customers. All vehicles start and end at the depot. For each pair of customers or customer and depot there is a cost. The cost denotes how much is costs a vehicle to drive from one customer to another. Every customer must be visited exactly ones. Additionally each customer demands a certain quantity of goods delivered (know as the customer demand). For the vehicles we have an upper limit on the amount of goods that can be carried (known as the capacity). In the most basic case all vehicles are of the same type and hence have the same capacity. The problem is now for a given scenario to plan routes for the vehicles in accordance with the mentioned constraints such that the cost accumulated on the routes, the xed costs (how much does it cost to maintain a vehicle) or a combination hereof is minimized. In the more general VRPTW each customer has a time window, and between all pairs of customers or a customer and the depot we have a travel time. The vehicles now have to comply with the additional constraint that servicing of the customers can only be started within the time windows of the customers. It is legal to arrive before a time window \opens" but the vehicle must wait and service will not start until the time window of the customer actually opens. For solving the problem exactly 4 general types of solution methods have evolved in the literature: dynamic programming, Dantzig-Wolfe (column generation), Lagrange decomposition and solving the classical model formulation directly. Presently the algorithms that uses Dantzig-Wolfe given the best results (Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis shows promising results for using the classical model formulation directly. In this Ph.D. project we have used the Dantzig-Wolfe method. In the Dantzig-Wolfe method the problem is split into two problems: a \master problem" and a \subproblem". The master problem is a relaxed set partitioning v vi problem that guarantees that each customer is visited exactly ones, while the subproblem is a shortest path problem with additional constraints (capacity and time window). Using the master problem the reduced costs are computed for each arc, and these costs are then used in the subproblem in order to generate routes from the depot and back to the depot again. The best (improving) routes are then returned to the master problem and entered into the relaxed set partitioning problem. As the set partitioning problem is relaxed by removing the integer constraints the solution is seldomly integral therefore the Dantzig-Wolfe method is embedded in a separation-based solution-technique. In this Ph.D. project we have been trying to exploit structural properties in order to speed up execution times, and we have been using parallel computers to be able to solve problems faster or solve larger problems. The thesis starts with a review of previous work within the eld of VRPTW both with respect to heuristic solution methods and exact (optimal) methods. Through a series of experimental tests we seek to dene and examine a number of structural characteristics. The rst series of tests examine the use of dividing time windows as the branching principle in the separation-based solution-technique. Instead of using the methods previously described in the literature for dividing a problem into smaller problems we use a methods developed for a variant of the VRPTW. The results are unfortunately not positive. Instead of dividing a problem into two smaller problems and try to solve these we can try to get an integer solution without having to branch. A cut is an inequality that separates the (non-integral) optimal solution from all the integer solutions. By nding and inserting cuts we can try to avoid branching. For the VRPTW Kohl has developed the 2-path cuts. In the separationalgorithm for detecting 2-path cuts a number of test are made. By structuring the order in which we try to generate cuts we achieved very positive results. In the Dantzig-Wolfe process a large number of columns may be generated, but a signicant fraction of the columns introduced will not be interesting with respect to the master problem. It is a priori not possible to determine which columns are attractive and which are not, but if a column does not become part of the basis of the relaxed set partitioning problem we consider it to be of no benet for the solution process. These columns are subsequently removed from the master problem. Experiments demonstrate a signicant cut of the running time. Positive results were also achieved by stopping the route-generation process prematurely in the case of time-consuming shortest path computations. Often this leads to stopping the shortest path subroutine in cases where the information (from the dual variables) leads to \bad" routes. The premature exit from the shortest path subroutine restricts the generation of \bad" routes signi cantly. This produces very good results and has made it possible to solve problem instances not solved to optimality before. The parallel algorithm is based upon the sequential Dantzig-Wolfe based algorithm developed earlier in the project. In an initial (sequential) phase unsolved problems are generated and when there are unsolved problems enough vii to start work on every processor the parallel solution phase is initiated. In the parallel phase each processor runs the sequential algorithm. To get a good workload a strategy based on balancing the load between neighbouring processors is implemented. The resulting algorithm is eÆcient and capable of attaining good speedup values. The loadbalancing strategy shows an even distribution of work among the processors. Due to the large demand for using the IBM SP2 parallel computer at UNIC it has unfortunately not be possible to run as many tests as we would have liked. We have although managed to solve one problem not solved before using our parallel algorithm.

Abstract. An exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the first time for several fairly large data sets from the literature, including Fisher’s 150 iris. Key words. classification and discrimination, cluster analysis, interior-point methods, combinatorial optimization

In this paper, the problem of bus driver scheduling is introduced, and some of the constraints and conditions existing in different user environments are presented. The way in which such conditions may affect solution methods is discussed. The development of driver scheduling by computer through the five prevous Workshops is presented, and the range of solution methods as evidenced by published papers is summarised. Particular attention is paid to the work presented at the later workshops, but papers published elsewhere are also introduced, and the authors draw on their own knowledge to augment the published material. Introduction In this paper we present a survey of computer approaches to transit driver scheduling, with particular emphasis on the state of the art as it existed at the time of the Montreal Workshop on Computer-Aided Scheduling of Public Transport in 1990 [5]. We start by outlining the driver scheduling problem and its variants. We then classify solution methods, and ...

We describe a methodology for finding near-optimal solutions to airline crew pairing problems. We use a dynamic column generation scheme to identify crew work schedules combined with a customized branch-and-bound procedure that allows column generation to be performed at each node of the search tree. Our approach provides an approximation to optimality since we only solve the column generation subproblems approximately and we do not necessarily consider all of the unexplored nodes in the search. We present computational results for both a research implementation and a production implementation of the algorithm on test problems from a major domestic carrier. We test the influence of various algorithmic design choices with the research implementation. These results were used to build a production implementation capable of finding good solutions to problem instances with over 2000 flight legs. June 23, 1997 0 This research has been supported by the following grants and contracts: NSF G...

Column generation has become a powerful tool in solving large scale integer programs. We argue that most of the often reported compatibility issues between pricing oracle and branching rules disappear when branching decisions are based on the reduction of the oracle's domain. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems. Our proposal opens the way for the development of branching rules adapted to the oracle structure and the coupling constraints.

The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS.

Column generation is an increasingly important technique for the solution of linear and integer programming problems. This paper describes a nested column generation approach for rostering problems in which the arc weights in a master shortest path problem are calculated by solving a second set of shortest path problems. This model exploits the natural objective function independence that occurs in many rostering problems. Results are presented for a nurse rostering example. 1. Introduction Operations Research literature is replete with examples of integer programming techniques being applied to rostering problems. These applications are increasingly making use of column generation techniques whereby the columns of the A matrix are not all explicitly enumerated, but instead are generated when needed, typically using shortest path approaches. (See, eg, Gamache [2] and Barnhart et. al. [1].) The development of efficient column generators for these problems underpins the successful appl...