Ellis L. Johnson

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.

MATCHING, EULER TOURS AND THE CHINESE POSTMAN

The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.

Solving Large-Scale Zero-One Linear Programming Problems

In this paper we report on the solution to optimality of 10 large-scale zero-one linear programming problems. All problem data come from real-world industrial applications and are characterized by sparse constraint matrices with rational data. About half of the sample problems have no apparent special structure; the remainder show structural characteristics that our computational procedures do not exploit directly. By todays standards, our methodology produced impressive computational results, particularly on sparse problems having no apparent special structure. The computational results on problems with up to 2,750 variables strongly confirm our hypothesis that a combination of problem preprocessing, cutting planes, and clever branch-and-bound techniques permit the optimization of sparse large-scale zero-one linear programming problems, even those with no apparent special structure, in reasonable computation times. Our results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem. To arrive at these conclusions, we designed an experimental computer system PIPX that uses the IBM linear programming system MPSX/370 and the IBM integer programming system MIP/370 as building blocks. The entire system is automatic and requires no manual intervention.

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.

Flight String Models for Aircraft Fleeting and Routing

Given a schedule of flight legs to be flown by an airline, the fleet assignment problem is to determine the minimum cost assignment of flights to aircraft types, called fleets, such that each scheduled flight is assigned to exactly one fleet, and the resulting assignment is feasible to fly given a limited number of aircraft in each fleet. Then the airline must determine a sequence of flights, or routes, to be flown by individual aircraft such that assigned flights are included in exactly one route, and all aircraft can be maintained as necessary. This is referred to as the aircraft routing problem. In this paper, we present a single model and solution approach to solve simultaneously the fleet assignment and aircraft routing problems. Our approach is robust in that it can capture costs associated with aircraft connections and complicating constraints such as maintenance requirements. By setting the number of fleets to one, our approach can be used to solve the aircraft routing problem alone. We show how to extend our model and solution approach to solve aircraft routing problems with additional constraints requiring equal aircraft utilization. With data provided by airlines, we provide computational results for the combined fleet assignment and aircraft routing problems without equal utilization requirements and for aircraft routing problems requiring equal aircraft utilization.

Facet of regular 0–1 polytopes

The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean functionβ on the one hand, and facets of the convex hullH of the zeros ofβ on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP ofβ, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets whenβ is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.

Some continuous functions related to corner polyhedra, II

Previous work on Gomorys corner polyhedra is extended to generate valid inequalities for any mixed integer program. The theory of a corresponding asymptotic problem is developed. It is shown how faces previously generated and those given here can be used to give valid inequalities for any integer program.

Min-cut clustering

We describe a decomposition framework and a column generation scheme for solving a min-cut clustering problem. The subproblem to generate additional columns is itself an NP-hard mixed integer programming problem. We discuss strong valid inequalities for the subproblem and describe some efficient solution strategies. Computational results on compiler construction problems are reported.

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.