Gabriele Sigismondi

The fleet assignment problem: solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.

Software section: MINTO, a mixed INTeger optimizer

MINTO is a software system that solves mixed-integer linear programs by a branch-and-bound algorithm with linear programming relaxations. It also provides automatic constraint classification, preprocessing, primal heuristics and constraint generation. Moreover, the user can enrich the basic algorithm by providing a variety of specialized application routines that can customize MINTO to achieve maximum efficiency for a problem class.

A column generation and partitioning approach for multi-commodity flow problems

Multi-commodity network flow problems, prevalent in transportation, production and communication systems, can be characterized by a set of commodities and an underlying network. The objective is to flow the commodities through the network at minimum cost without exceeding arc capacities. In this paper, we present a partitioning solution procedure for large-scale multi-commodity flow problems with many commodities, such as those encountered in the telecommunications industry. Using a cycle-based multi-commodity formulation and column generation techniques, we solve a series of reduced-size linear programs in which a large number of constraints are relaxed. Each solution to a reduced-size problem is an improved basic dual feasible solution to the original problem and, after a finite number of steps, an optimal multi-commodity flow solution is determined. Computational experience is gained in solving randomly generated test problems and message routing problems in the communications industry. The tests show that the procedure solves large-scale multi-commodity flow problems significantly faster than existing linear programming or column generation solution procedures.

Formulating a Mixed Integer Programming Problem to Improve Solvability

A standard formulation of a real-world distribution problem could not be solved, even for a good solution, by a commercial mixed integer programming code. However, after reformulating it by reducing the number of 0-1 variables and tightening the linear programming relaxation, an optimal solution could be found efficiently. The purpose of this paper is to demonstrate, with a real application, the practical importance of the need for good formulations in solving mixed integer programming problems.

A Characterization of Lifted-Cover Facets of 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. We extend the results of Balas [1] and Balas and Zemel [2] by giving a set of inequalities that determines the lifting coefficients of facet-defining inequalities for any ordering of the variables to be lifted. We further extend these results to obtain facet-defining inequalities for the 0–1 knapsack problem with mutually exclusive generalized upper bound (GUB) constraints.