J. E. Beasley

A genetic algorithm for the set covering problem

Abstract In this paper we present a genetic algorithm-based heuristic for non-unicost set covering problems. We propose several modifications to the basic genetic procedures including a new fitness-based crossover operator (fusion), a variable mutation rate and a heuristic feasibility operator tailored specifically for the set covering problem. The performance of our algorithm was evaluated on a large set of randomly generated problems. Computational results showed that the genetic algorithm-based heuristic is capable of producing high-quality solutions.

A Genetic Algorithm for the Multidimensional Knapsack Problem

In this paper we present a heuristic based upon genetic algorithms for the multidimensional knapsack problem. A heuristic operator which utilises problem-specific knowledge is incorporated into the standard genetic algorithm approach. Computational results show that the genetic algorithm heuristic is capable of obtaining high-quality solutions for problems of various characteristics, whilst requiring only a modest amount of computational effort. Computational results also show that the genetic algorithm heuristic gives superior quality solutions to a number of other heuristics.

Heuristics for cardinality constrained portfolio optimisation

In this paper we consider the problem of finding the efficient frontier associated with the standard mean-variance portfolio optimisation model. We extend the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets, and to impose limits on the proportion of the portfolio held in a given asset (if any of the asset is held). We illustrate the differences that arise in the shape of this efficient frontier when such constraints are present. We present three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing for finding the cardinality constrained efficient frontier. Computational results are presented for five data sets involving up to 225 assets.

An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure

We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.

A genetic algorithm for the generalised assignment problem

Abstract In this paper we present a genetic algorithm (GA)-based heuristic for solving the generalised assignment problem. The generalised assignment problem is the problem of finding the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent, subject to an agents capacity. In addition to the standard GA procedures, our GA heuristic incorporates a problem-specific coding of a solution structure, a fitness-unfitness pair evaluation function and a local improvement procedure. The performance of our algorithm is evaluated on 84 standard test problems of various sizes ranging from 75 to 4000 decision variables. Computational results show that the genetic algorithm heuristic is able to find optimal and near optimal solutions that are on average less than 0.01 % from optimality. The performance of our heuristic also compares favourably to all other existing heuristic algorithms in terms of solution quality.

Scheduling Aircraft Landings--The Static Case

In this paper, we consider the problem of scheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time for each plane such that each plane lands within a predetermined time window and that separation criteria between the landing of a plane and the landing of all successive planes are respected. We present a mixed-integer zero--one formulation of the problem for the single runway case and extend it to the multiple runway case. We strengthen the linear programming relaxations of these formulations by introducing additional constraints. Throughout, we discuss how our formulations can be used to model a number of issues (choice of objective function, precedence restrictions, restricting the number of landings in a given time period, runway workload balancing) commonly encountered in practice. The problem is solved optimally using linear programming-based tree search. We also present an effective heuristic algorithm for the problem. Computational results for both the heuristic and the optimal algorithm are presented for a number of test problems involving up to 50 planes and four runways.

Lagrangean heuristics for location problems

Abstract In this paper we present a framework for developing Lagrangean heuristics (heuristics based upon Lagrangean relaxation and subgradient optimisation) with respect to location problems. Computational results are given for four different location problems: p-median, uncapacitated warehouse location, capacitated warehouse location and capacitated warehouse location with single source constraints. These results indicate that the framework presented in this paper is robust, i.e. it gives good quality solutions for each of these different location problems.

The period routing problem

In this paper we present heuristic algorithms for the period vehicle routing problem, the problem of designing vehicle routes to meet required service levels for customers and minimize distribution costs over a given several-day period of time. These heuristic algorithms are based on an initial choice of customer delivery days which meet the service level requirements, followed by an interchange procedure in an attempt to minimize distribution costs. The heuristic algorithms represent distribution costs by replacing the vehicle routing problem for each day of the period by (I) a median problem and (II) a traveling salesman problem. Computational results and comparisons are given for the algorithms, based on test problems derived from the literature with up to 126 customers. The largest of these problems is the one given and solved by Russell and Igo. The solution obtained for this problem by the heuristic algorithms shows an improvement of 13% over the previous best solution. (Author/TRRL)

An algorithm for the resource constrained shortest path problem

In this paper we examine an integer programming formulation of the resource constrained shortest path problem. This is the problem of a traveller with a budget of various resources who has to reach a given destination as quickly as possible within the resource constraints imposed by his budget. A lagrangean relaxation of the integer programming formulation of the problem into a minimum cost network flow problem (which in certain circumstances reduces to an unconstrained shortest path problem) is developed which provides a lower bound for use in a tree search procedure. Problem reduction tests based on both the original problem and this lagrangean relaxation are given. Computational results are presented for the solution of problems involving up to 500 vertices, 5000 arcs, and 10 resources.

Comparing university departments

In this paper we present a quantitative model for comparing university departments concerned with the same discipline. This model is based upon ideas drawn from data envelopment analysis. Computational results are given for chemistry and physics departments in the United Kingdom.