Interactive approaches to the solution of a class of combinatorial problems

Abstract

This thesis considers the usefulness of interaction between a human and a powerful computer in attempting to solve a class of discrete optimization problems. Some typical problems are described in chapters 1 and 2 and the effectiveness of their exact solution by existing methods is assessed. Chapter 3 presents some heuristic techniques which produce good approximate solutions and the value of such methods is discussed. An alternative approach, that of providing a mechanism for manmachine interaction is proposed in chapter 4. A system for providing easy access to a range of algorithmic and heuristic techniques is described. The system, named IMPACT, was implemented by the author and its many features include the interruption, interrogation, adjustment and resumption of a process or algorithm. Some novel interactive tree-manipulation techniques and their usage are introduced in chapter 5. This chapter also describes extensions to certain other heuristics in order to improve their power when used interactively. Throughout the thesis a job-shop scheduling problem serves as a useful vehicle for illustrating ideas. This problem was investigated extensively and chapter 6 is devoted to the topic. The idea of a critical path of jobs through machines is introduced together with the slack time of a job upon a machine under a particular schedule. Branch-and-bound approaches to the problem have been proposed in the past. The performance of such an approach has been substantially improved, as is shown by new results. The improvement stems from two sources both of which were discovered interactively; i) a different branching procedure designed to exploit features of the job-shop scheduling problem, and ii) more realistic lower bounds than those originally proposed. The final chapter discusses the generality of the approach and illustrates the extendability of IMPACT. Other discrete optimization problems are discussed briefly and a branch-andbound formulation to one of them, an assignment problem~ is presented. An interactive approach by other authors to the travelling salesman problem is reviewed and features similar to those experienced in the job-shop scheduling investigation are remarked upon. To conclude, the advantages to be gained from an interactive approach are discussed.