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."

Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NP-complete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and lower bounding (also across components), elaborate and dynamically chosen bid-ordering heuristics, and a host of structural observations. CABOB attempts to capture structure in any instance without making assumptions about the instance distribution. Experiments against the fastest prior algorithm, CPLEX 8.0, show that CABOB is often faster, seldom drastically slower, and in many cases drastically faster—especially in cases with structure. CABOB’s search runs in linear space and has significantly better anytime performance than CPLEX. We also uncover interesting aspects of the problem itself. First, problems with short bids, which were hard for the first generation of specialized algorithms, are easy. Second, almost all of the CATS distributions are easy, and the run time is virtually unaffected by the number of goods. Third, we test several random restart strategies, showing that they do not help on this problem—the run-time distribution does not have a heavy tail.

This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.

Crew management is concerned with building the work schedules of crews needed to cover a planned timetable. This is a well-known problem in Operations Research and has been historically associated with airlines and mass-transit companies. More recently, railway applications have also come on the scene, especially in Europe. In practice, the overall crew management problem is decomposed into two subproblems, called crew scheduling and crew rostering. In this paper, we give an outline of different ways of modeling the two subproblems and possible solution methods. Two main solution approaches are illustrated for real-world applications. In particular we discuss in some detail the solution techniques currently adopted at the Italian railway company, Ferrovie dello Stato SpA, for solving crew scheduling and rostering problems.

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50–80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities. 1

Given a finite ground set, a set of subsets and costs on the subsets, the set partitioning problem is to find a minimum cost partition of the ground set. Many combinatorial optimization problems can be formulated as set partitioning problems. We present an approximation algorithm that produces high quality solutions in an acceptable amount of computation time. The algorithm is iterative and combines problem size reduction techniques, such as logical implications derived from feasibility and optimality conditions and reduced cost fixing, with a primal heuristic based on cost perturbations embedded in a Lagrangian dual framework. Computational experiments illustrate the effectiveness of the approximation algorithm. Keywords: set partitioning, preprocessing, linear programming, Lagrangian dual September 1995 1 Introduction Given a finite ground set, a set of subsets and costs on the subsets, the set partitioning problem is to find a minimum cost partition of the ground set. Let A be an...

We consider a pickup and delivery vehicle routing problem commonly encountered in real-world logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the driver's work rules prescribed by the Department of Transportation which specify legal working hours of ...

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.