François Vanderbeck

On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm

Dantzig-Wolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound. The reformulation gives rise to an integer master problem, whose typically large number of variables is dealt with implicitly by using an integer programming column generation procedure, also known as branch-and-price algorithm. There is a large class of integer programs that are well suited for this solution technique. In this paper, we propose to base the Dantzig-Wolfe decomposition of an integer program on the discretization of the integer polyhedron associated with a subsystem of constraints (as opposed to its convexification). This allows us to formulate the integrality restriction directly on the master variables and sets a theoretical framework for dealing with specific issues such as branching or the introduction of cutting planes in the master. We discuss specific branching schemes and their effect on the structure of the column generation subproblem. We give theoretical bounds on the complexity of the separation process and the extent of the modifications to the column generation subproblem. Our computational tests on the cutting stock problem and a generalisation--the cutting strip problem--show that, in practice, all fractional solutions can be eliminated using branching rules that preserve the tractability of the subproblem, but there is a trade-off. between branching efficiency and subproblem tractability.

An exact algorithm for IP column generation

An exact column generation algorithm for integer programs with a large (implicit) number of columns is presented. The family of problems that can be treated includes not only standard partitioning problems such as bin packing and certain vehicle routing problems in which the columns generated have 0-1 compenents and a right-hand side vector of 1s, but also the cutting stock problem in which the columns and right-hand side are nonnegative integer vectors. We develop a combined branching and subproblem modification scheme that generalizes existing approaches, and also describe the use of lower bounds to reduce tailing-off effects.

Computational study of a column generation algorithm for bin packing and cutting stock problems

Abstract.This paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem. The main focus of the research is to study the extend to which standard branch-and-bound enhancement features such as variable fixing, the tightening of the formulation with cutting planes, early branching, and rounding heuristics can be usefully incorporated in a branch-and-price algorithm. We review and compare lower bounds for the cutting stock problem. We propose a pseudo-polynomial heuristic. We discuss the implementation of the important features of the integer programming column generation algorithm and, in particular, the implementation of the branching scheme. Our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems.

Implementing Mixed Integer Column Generation

We review the main issues that arise when implementing a column generation approach to solve a mixed integer program: setting-up the Dantzig-Wolfe reformulation, adapting standard MIP techniques to the context of column generation (branching, preprocessing, primal heuristics), and dealing with issues specific to column generation (initialization, stabilization, column management strategies). The description of the different features is done in generic terms to emphasize their applicability across problems. More hand-on experiences are reported in the literature in application specific context, f.i., see Desaulniers et al. (2001) for vehicle routing and crew scheduling applications. This paper summarizes recent work in the field, in particular that of Vanderbeck (2002, 2003).

Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem

The cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste. Because changing over from one cutting pattern to another involves significant setups, an auxiliary problem is to minimise the number of different patterns that are used. The pattern minimisation problem is significantly more complex, but it is of great practical importance. In this paper, we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns, each of which corresponds to selecting a fixed number of copies of a specific cutting pattern. The integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems. At each node of the branch-and-bound tree, the linear programming relaxation of our formulation is made tighter by adding super-additive inequalities. Branching rules are presented that yield a balanced tree. Incumbent solutions are obtained using a rounding heuristic. The resulting branch-and-price-and-cut procedure is used to produce optimal or approximately optimal solutions for a set of real-life problems.

Comparison of bundle and classical column generation

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

A generic view of Dantzig-Wolfe decomposition in mixed integer programming

The Dantzig-Wolfe reformulation principle is presented based on the concept of generating sets. The use of generating sets allows for an easy extension to mixed integer programming. Moreover, it provides a unifying framework for viewing various column generation practices, such as relaxing or tightening the column generation subproblem and introducing stabilization techniques.

Column generation based heuristic for tactical planning in multi-period vehicle routing

The periodic vehicle routing problem (PVRP) consists in establishing a planning of visits to clients over a given time horizon so as to satisfy some service level while optimizing the routes used in each time period. The tactical planning model considered here restricts its attention to scheduling visits and assigning them to vehicles while leaving sequencing decisions for an underlying operational model. The objective is twofold: to optimize regional compactness of the routes in a desire to specialize routes to restricted geographical area and to balance the workload evenly between vehicles. Approximate solutions are constructed using a truncated column generation procedure followed by a rounding heuristic. This mathematical programming based procedure can deal with problems with 50–80 customers over five working days which is the range of size of most PVRP instances treated in the literature with meta-heuristics. The paper highlights the importance of alternative optimization criteria not accounted for in standard operational models and provides insights on the implementation of a column generation based rounding heuristic.

A Nested Decomposition Approach to a Three-Stage, Two-Dimensional Cutting-Stock Problem

We consider the cutting of rectangular order pieces into stock pieces of specified width and length. The cutting process involves three stages of orthogonal guillotine cutting: Stock pieces are cut into sections that are cut into slits that are cut into order pieces. Restrictions imposed on the cutting process make the combinatorial structure of the problem more complex, but limit the scope of solution space. The objective of the problem is mainly to minimize waste, but our model also accounts for other issues such as aging stock pieces, urgent or optional orders, and fixed setup costs. Our solution approach involves a nested decomposition of the problem and the recursive use of the column-generation technique: We use a column-generation formulation of the problem (Gilmore and Gomory 1965) and the cutting-pattern--generation subproblem is itself solved using a column-generation algorithm. LP-based lower bounds on the minimum cost are computed and, by rounding the LP solution, a feasible solution and associated upper bound is obtained. This approach could in principle be used in a branch-and-bound search to solve the problem to optimality. We report computational results for industrial instances. The algorithm is being used in industry as a production-planning tool.

Branching in branch-and-price: a generic scheme

Developing a branching scheme that is compatible with the column generation procedure can be challenging. Application specific and generic schemes have been proposed in the literature, but they have their drawbacks. One generic scheme is to implement standard branching in the space of the compact formulation to which the Dantzig-Wolfe reformulation was applied. However, in the presence of multiple identical subsystems, the mapping to the original variable space typically induces symmetries. An alternative, in an application specific context, can be to expand the compact formulation to offer a wider choice of branching variables. Other existing generic schemes for use in branch-and-price imply modifications to the pricing problem. This is a concern because the pricing oracle on which the method relies might become obsolete beyond the root node. This paper presents a generic branching scheme in which the pricing oracle of the root node remains of use after branching (assuming that the pricing oracle can handle bounds on the subproblem variables). The scheme does not require the use of an extended formulation of the original problem. It proceeds by recursively partitioning the subproblem solution set. Branching constraints are enforced in the pricing problem instead of being dualized via Lagrangian relaxation, and the pricing problem is solved by a limited number of calls to the pricing oracle. This generic scheme builds on previously proposed approaches and unifies them. We illustrate its use on the cutting stock and bin packing problems. This is the first branch-and-price algorithm capable of solving such problems to integrality without modifying the subproblem or expanding its variable space.