Mathematicians love the "travelling salesman problem" because it is easy to state - find the shortest route between a specified number of cities - but too complex to solve. If there are 15 cities, for example, there are billions of possible routes.

Yet supply chain managers of large companies face even more complex versions of this mathematical curiosity every day.

Air Liquide, the French industrial gases group, makes liquid and gaseous oxygen, nitrogen, argon, carbon dioxide and other gases, which it delivers in cryogenic tankers to customer sites. In the US alone the company serves 10,000 sites, from more than 200 sources, through 20 depots, using 300 trailers and 100 tractor units.

It is an extremely difficult scheduling and distribution problem, to which the company has found an unusual solution based on complexity science and on tracking the behaviour of "software ants".

Charles Harper, director of pipeline and supply operations, primary production and manufacturing at Air Liquide's US business, wanted a system to help the logistics analysts and schedulers who operated the supply chain to make better decisions. "We talked with several supply chain optimisation vendors," says Mr Harper, "but we quickly realised they couldn't cope with the complexity of our problem. Once you get past the sales pitch, you realise they're selling a package you must conform to. They'll tweak it a bit for you - but the basic engine never changes."

BiosGroup seemed a risky choice. The five-year-old company, co-founded by Stuart Kauffman, uses exotic approaches based on complexity science, the study of "non-linear" or "complex adaptive" systems. Such systems are common in nature and business and cannot be controlled by outside means - but they can be under the control of their own self-organising properties.

"It was a different way of looking at the problem," Mr Harper says. "We read Stu Kauffman's books, talked to BiosGroup scientists and realised we needed a non-linear approach. You never get to talk to the chief developers with other vendors."

A team at BiosGroup, led by David Thompson, chief project scientist, were given the go-ahead to use "agent" and "ant-based" algorithms to develop a decision support system for Air Liquide.

Agents were created, to represent all supply chain components - production plant, customers, trucks, drivers and so on - and an ant-based model was used for route planning. Software ants explored a space consisting of all possible routes, laying pheromone trails (like real ants) and following simple rules, such as "follow strong pheromone trails".

From time to time, equivalents of branches and rocks were tossed among the ants to represent truck break-downs, plant shutdowns, or changes in demand or prices. Like real ants, the software ants soon found routes around the obstacles.

The "engine" derived from these simulations provides logistics analysts and plant schedulers with daily recommendations. It has valuable qualities that are not shared by conventional software. It is robust in the sense that if one part fails, the system as a whole can still perform; it is self-organising in that it requires no central control; and because agent and ant models generate many near-optimal solutions, rescheduling is easy.

"The key requirement," says Mr Thompson, "was getting flexibility into the reinforcement algorithms to ensure that local decisions reflected feedback from the whole system. The analysts knew their customers well. We captured the rules they use intuitively and tested them. Some were fine but others, such as 'trucks must always be sent out full' and 'trucks must always come back empty', turned out to be inefficient."

Mr Thompson says the engine could never take over completely from the analysts and schedulers, because the number of rules and exceptions is very great. "But if the engine can handle 90 per cent of the decisions, the analysts and schedulers can concentrate on the other 10 per cent."

Clarke Hayes, Air Liquide's project leader, was careful to make it clear that intuition remained an integral part of the system.

The company is still analysing the results of the system but it says there have been substantial benefits in cost savings and customer satisfaction.

"Most of the cost in a supply chain is in the interfaces between departments and people," says Mr Harper. "We got a completely customised solution from Bios, covering all of those interfaces, for less than we would have paid for a conventional package with a few tweaks."

Mr Harper is already discussing a second project to help Air Liquide make money from selling elect-ricity.

"We produce more power than we consume, so our 10,000 storage tanks of liquid gas are (in effect) electrical capacitors. We think that we could make another Dollars 5m (Pounds 3.4m) a year of profit if we had an engine to help us work out when it is economic and safe to sell power," he says. "That's Phase two. Bios also has systems for analysing risk. That could be phase three."

Meanwhile, Mr Thompson says, complexity-based systems are gradually being adopted by companies.

Potential applications include genetic algorithms, which mimic natural selection, and systems that simulate colonies of insects. Complexity-based business tools are also being used for battlefield simulations, to route traffic through telephone networks, to analyse financial and operational risk and to provide decision support to fund managers.

Copyright: The Financial Times Limited 1995-2002