Les G. Proll

An algorithm for determining all extreme points of a convex polytope

An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.

An improved ILP system for driver scheduling

Mathematical programming approaches to driver scheduling have been reported at many previous workshops and have become the dominant approach to the problem. However the problem frequently is too large for mathematical programming to be able to guarantee an optimal schedule. TRACS II, developed at the University of Leeds, is one such mathematical programming-based scheduling system. Several improvements and alternative solution methods have now been incorporated into the mathematical programming component of the TRACS II system, including a column generation technique which implicitly considers many more valid shifts than standard linear programming approaches. All improvements and alternative strategies have been implemented into the mathematical programming component of TRACS II to allow different solution methods to be used where necessary, and to solve larger problems in a single pass, as well as to produce better solutions. Comparative results on real-world problems are presented.

Integer Linear Programming and Constraint Programming Approaches to a Template Design Problem

We describe a design problem arising in the color printing industry and discuss a number of integer linear programming and constraint programming approaches to its solution. Despite the apparent simplicity of the problem it presents a challenge for both approaches. We present results for three typical cases and show that the constraint programming approach provides better results, although in some cases after considerable handcrafting. We also show that the results obtained by constraint programming can be improved by a simple goal programming model.

TRACS II: a hybrid IP/heuristic driver scheduling system for public transport

We discuss the driver scheduling problem in public transport and describe a combined integer linear programming/heuristic approach to its solution. The approach has been applied successfully in many operational and planning scenarios. Recent developments in the algorithms used allow the solution of very large bus and rail problems.

Experiences with a Flexible Driver Scheduler

We present a flexible user-driven ILP tool for the optimisation component of the TRACS II driver scheduling system. The system allows the user to select from a number of objective functions and to drive the LP relaxation through one of a range of optimisation processes. As a default we provide a Sherali objective which minimises the number of shifts, and within that yields the least cost. The default method of solving the LP relaxation is by specialised primal column generation for larger problems, and by dual steepest edge for smaller ones. The LP is capable of working with over 200, 000 previously generated shifts. Once the relaxed LP has been solved, we reduce the size of the problem, and enter a specialised branch and bound procedure. Using real data we show how, by being able to handle larger problems, our system reduces the need for problem decomposition and can produce better schedules.

A flexible system for scheduling drivers

A substantial part of the operating costs of public transport is attributable to drivers, whose efficient use therefore is important. The compilation of optimal work packages is difficult, being NP-hard. In practice, algorithmic advances and enhanced computing power have led to significant progress in achieving better schedules. However, differences in labor practices among modes of transport and operating companies make production of a truly general system with acceptable performance a difficult proposition. TRACS II has overcome these difficulties, being used with success by a substantial number of bus and train operators. Many theoretical aspects of the system have been published previously. This paper shows for the first time how theory and practice have been brought together, explaining the many features which have been added to the algorithmic kernel to provide a user-friendly and adaptable system designed to provide maximum flexibility in practice. We discuss the extent to which users have been involved in system development, leading to many practical successes, and we summarize some recent achievements.

An improved vertex enumeration algorithm

Abstract The main computational burden in pivoting methods for determining all vertices of a convex polytope appears to be in testing pairs of vertices for adjacency. We show how, in the Dyer-Proll algorithm, this burden can be reduced by a new labelling of the search tree and by a mechanism for removing redundant branches. We also introduce an implementation strategy, the barred pivot strategy, which further improves the algorithms performance.

A dual strategy for solving the linear programming relaxation of a driver scheduling system

A Mathematical Programming model of a driver scheduling system is described. This consists of set covering and partitioning constraints, possibly user-supplied side constraints, and two pre-emptively ordered objectives. The previous solution strategy addressed the two objectives using separate Primal Simplex optimisations; a new strategy uses a single weighted objective function and a Dual Simplex algorithm initiated by a specially developed heuristic. Computational results are reported.

Mathematical programming and the sensitivity of multi-criteria decisions

We report on the current state of a project whose aim is to implement a framework for sensitivity analysis in Multi-Criteria Decision Making (MCDM). The framework is largely based on mathematical programming. Due to the potentially large number and nature of the mathematical programmes, it is far from trivial to provide solutions to all of them in acceptable computing times. The challenge is even greater when we recognize that much decision analysis is performed in the context of decision conferences where any sensitivity analysis needs to be conducted in near real time (preferably) on a PC. We present a parallel processing approach to this challenge and point to some of the difficulties still to be resolved. Preliminary results obtained on a network of transputers are discussed.

ILP approaches to the blockmodel problem

Blockmodelling is a method for identifying structural similarities or equivalences between elements which has applications in a variety of contexts, including multiattribute performance assessment. One criterion for forming blocks results in a difficult non-linear integer programme. We give several integer linear programming formulations of this problem and provide comparative computational results. We show that methods of reducing symmetry proposed by Sherali and Smith are not effective in this case and propose an iterative approach in which the size of the problem is reduced.