AbstractMany large practical combinatorial problems await methods of solution, especially those involving the scheduling of hundreds or thousands of activities. The travelling salesman problem is archetypal and has attracted much attention, yet published rigorous methods can only be applied to problems encompassing tens of cities and approximate methods appear only to have been applied to problems of up to 105 cities. None of these methods offers scope for extension to much larger problems because of rapidly increasing computing requirements.Some approximate, sequential methods and the results of applying them to thirty problems of between 20 and 500 cities are described. In addition, the results of applying one method to ten 2500 city problems are reported; the solutions produced were, on average, 1·28 times the corresponding lower bound. It appears that satisfactory solutions to some large practical problems may be obtained by developing suitable sequential methods.