Pamela H. Vance

Branch-And-Price: Column Generation for Solving Huge Integer Programs

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 branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems

We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origin destination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined column and- cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

Solving binary cutting stock problems by column generation and branch-and-bound

We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.

Airline Crew Scheduling

An airline must cover each flight leg with a full complement of cabin crew in a manner consistent with safety regulations and award requirements. Methods are investigated for solving the set partitioning and covering problem. A test example illustrates the problem and the use of heuristics. The Study Group achieved an understanding of the problem and a plan for further work.

Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem

We compare two branch-and-price approaches for the cutting stock problem. Each algorithm is based on a different integer programming formulation of the column generation master problem. One formulation results in a master problem with 0–1 integer variables while the other has general integer variables. Both algorithms employ column generation for solving LP relaxations at each node of a branch-and-bound tree to obtain optimal integer solutions. These different formulations yield the same column generation subproblem, but require different branch-and-bound approaches. Computational results for both real and randomly generated test problems are presented.

Optimal facility layout design11This study was supported in part by an Auburn College of Engineering infrastructure award made to Drs. Meller and Vance, and NSF CAREER Grants DMII 9623605 and DMII 9502502.

The facility layout problem (FLP) is a fundamental optimization problem encountered in many manufacturing and service organizations. Montreuil introduced a mixed integer programming (MIP) model for FLP that has been used as the basis for several rounding heuristics. However, no further attempt has been made to solve this MIP optimally. In fact, though this MIP only has 2n(n-1) 0-1 variables, it is very difficult to solve even for instances with n~5 departments. In this paper we reformulate Montreuils model by redefining his binary variables and tightening the department area constraints. Based on the acyclic subgraph structure underlying our model, we propose some general classes of valid inequalities. Using these inequalities in a branch-and-bound algorithm, we have been able to moderately increase the range of solvable problems. We are, however, still unable to solve problems large enough to be of practical interest. The disjunctive constraint structure underlying our FLP model is common to several other ordering/arrangement problems; e.g., circuit layout design, multi-dimensional orthogonal packing and multiple resource constrained scheduling problems. Thus, a better understanding of the polyhedral structure of this difficult class of MIPs would be valuable for a number of applications.

Railroad Blocking: A Network Design Application

In this study, we formulate the railroad blocking problems as a network design problem with maximum degree and flow constraints on the nodes and propose a heuristic Lagrangian relaxation approach to solve the problem. The newapproach decomposes the complicated mixed integer programming problem into two simple subproblems so that the storage requirement and computational effort are greatly reduced. A set of inequalities are added to one subproblem to tighten the lower bounds and facilitate generating feasible solutions. Subgradient optimization is used to solve the Lagrangian dual. An advanced dual feasible solution is generated to speed up the convergence of the subgradient method. The model is tested on blocking problems from a major railroad, and the results show that the blocking plans generated have the potential to reduce the railroads operating costs by millions of dollars annually.

Constructing Railroad Blocking Plans to Minimize Handling Costs

On major domestic railroads, a typical general merchandise shipment may pass through many classification yards on its route from origin to destination. At these yards, the incoming traffic, which may consist of a number of individual shipments, is reclassified (sorted and grouped together) to be placed on outgoing trains. Each reclassification incurs costs due to handling and delay. To prevent shipments from being reclassified at every yard they pass through, several shipments may be grouped together to form a block. A block has associated with an origin destination pair that may or may not be the origin or destination of any of the individual cars contained in the block. The objective of the railroad blocking problem is to choose which blocks to build at each yard and to assign sequences of blocks to deliver each shipment to minimize total mileage, handling, and delay costs. We model the railroad blocking problem as a network design problem in which yards are represented by nodes and blocks by arcs. Our model is intended as a strategic decision-making tool. We develop a column generation, branch-and-bound algorithm in which attractive paths for each shipment are generated by solving a shortest path problem. Our solution approach is unique in constraining the classification resources of each yard and simultaneously solving for different priority classes of shipments. We implement our algorithm and find near-optimal solutions in about one hour for the blocking problem of a large domestic railroad, in which the paths that shipments may take in the physical network are restricted. The resulting network design problem has 150 nodes, 1300 commodities, and 6800 possible arcs (blocks). We test the robustness of our solution on 19 test instances that are variations of the data for the real-world problems. If shipments are restricted to following one of a limited number of paths in the rail network, then, in four hours or less, our algorithm finds solutions within 0.4% of optimal for all test cases. Furthermore, the solutions obtained are no more than 3.9% from optimal even if all possible paths are allowed.

Lifted cover facets of the 0-1 knapsack polytope with GUB constraints

Facet-defining inequalities lifted from minimal covers are used as strong cutting planes in algorithms for solving 0-1 integer programming problems. In this paper we extend the result of Balas and Zemel by giving a set of inequalities that determines the lifting coefficients of facet-defining inequalities of the 0-1 knapsack polytope for any ordering of the variables to be lifted. We further generalize the result to obtain facet-defining inequalities for the 0-1 knapsack problem with generalized upper bound constraints.

Integer multicommodity flow problems

We present a column generation model and solution approach for large integer multicommodity flow problems. We solve the model using branch-and-bound, with bounds provided by linear programs at each node of the branch-and-bound tree. Since the model contains one variable for each origin-destination path, for every commodity, the linear programming relaxation is solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. Methods for speeding up the solution of the linear program are presented. Also, we devise new branching rules that allow columns to be generated efficiently at each node of the branch-and-bound tree. Computational results are presented for a set of test problems arising from a transportation application.