27: Beware of optimisation

The idea of ‘optimisation’ is basic to lots of things we do and to how we think. When driving from A to B, what is the optimum route? When we learn calculus for the first time, we quickly come to grips with the maximisation and minimisation of functions. This is professionalised within operational research. If you own a transport business, you have to plan a daily schedule of collections and deliveries. How do you allocate your fleet to miminise costs and hence to maximise profits for the day? In this case, the mathematics and the associated computer programmes exist and are well known – they will solve the problem for you. You have the information and you can control what happens. But suppose now that you are an economist and you want to describe, theorise about, or model human behaviour. Suppose you want to investigate the economics of the journey to work. This is another kind of scheduling problem except that in this case it involves a large number of individual decision makers. If we turn to the micro economics text books, we find the answer: define a utility function for individuals, and each can then maximise. In this case we run into problems: does each individual have all the relevant information? Does the economist have this? For the individual, all options need to be available on possible journeys – perfect information. The impossibility of this led Herbert Simon to the powerful concept of ‘satisficing’ rather than ‘maximising’. This was brilliant and shifts the modelling task to a probabilistic one (as well as being a realistic description of human behaviour – isn’t it what we all do?). Of course, economists responded to this too and associated probability distributions with the utility functions. This is more difficult to build into the basic economics’ text books, however. (And for this problem, there is an associated issue: space. The economists’ toolkit is usually seen as having micro or macro dimensions but when space needs to be added, we might think of the resulting scale as being ‘meso’. More adaptation needed.)

So the lesson to heed at this stage is that while ‘optimisation’ is a powerful concept and tool, it should be used with care and should be ‘blurred’ via the introduction of probability distributions – which makes everything more messy – when appropriate. Hence, we should ‘beware’. A related field to explore in this respect is the very fashionable agent-based models (ABM). If individuals in an ABM are behaving according to ‘rules’, does the specification of these rules incorporate the necessary blurring? I know this can be done, and have done it, but is it always done? I suspect not.

There is a temptation to stick to simpler forms of optimisation because the associated mathematics and computer software is so attractive. This is particularly true of linear programming problems like the transport owner’s scheduling problem. A good example is the algorithm for the shortest path through a network which Dijkstra discovered in the 1950s. A great thing to have. In the early days of transport modelling this provided the basis for assigning origin-destination flows to networks – essentially assuming that all travellers

took the best route. Again, this proved too simple and eventually it became possible, from the early 1970s, though a shade more difficult, to calculate second best and third best routes, even the kth best route – and then the trips could allocated probabilistically. And then we have to recognise that much of the world is nonlinear. The mathematics of nonlinear optimisation is more tricky but huge progress has been made – and indeed one of the core methods dates back to Lagrange in the Eighteenth Century.

I have been lucky with my own work in all these respects because entropy-maximising is a branch of nonlinear optimisation which actually offers optimum blurring. It is possible to take a traditional economic model and to turn it into an optimally blurred and hence more realistic model by these means. Indeed, there is one remarkable mathematical result: that the nonlinear version of an equivalent of the scheduling problem transforms into the linear problem when one of the parameters becomes infinite.

Conclusion: different kinds of optimisation methods should be in the toolkit, but we should be wary about matching uses with reality!

Alan Wilson

November 2015

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