Complexity and simulations, incompatible?

As practical experience suggets, Monte Carlo simulations are often associated with an increase in complexity. We find that instruments such as Monte Carlo simulations can neither increase nor reduce complexity.   

The concept of complexity fills thousands of books. The search on Google alone brought about 5 million hits (called in February 2017). In general, the behaviour of a system or model, whose components can interact with one another in various ways, is called complex. The multiplicity and interdependence between the components is therefore essential, whereby both dimensions do not necessarily have to occur at the same time. Such a constellation can thus also be measured, whereby the "variety" is being used as metric. Variety is defined as the number of possible, distinguishable states that a system may have.

 

According to the concept of complexity in theoretical computer science, the problem of the travelling salesman (TSP) can be used as an example. The problem is easy to explain and also easy to solve. There is a salesman who visits n cities and returns to his home town. What is the shortest way? With three cities (A, B, C), where A is the home town and the distance between the cities is assumed to be asymmetric, there are two different ways, with four cities the possibility increase to six. The solution to the problem is simply to list all combinations and select the one that has the shortest path. So easy. But what makes the simple problem a complex system? It is the sheer number of alternatives to be evaluated with increasing number "n", which makes it difficult to solve it in a reasonable time. Formally, (n-1)! (factorial) alternatives have to be evaluated. With a small number of cities we would be able to speak of a "simple" system, but a round trip which visits all the cities of the world would be defined a complex system. In practice, the solution to the TSP is often accomplished with heuristics as an approximation, such as the "nearest neighbour" procedure, in which from any starting point the nearest city, which has not yet been visited is passed until the last one, from which we return to the starting point. With this simple method, even in the case of a large number n, a solution is found in a reasonable time. However, the complexity has not been reduced (the "variety" would still be (n-1)!). Basically, you have excluded only certain information by using a heuristic.

 

In the case of systems that are characterized by uncertainty, decision-makers face a complex problem since many different alternatives, each with different consequences and coupling effects, have to be considered. The concept of single-value planning, which is prominent in business planning, reduces all possible future alternatives to one manifestation. Similarly to the TSP problem, certain information is simply ignored during the planning process. Concretely, fixing the variables to a concrete number ignores the uncertain events. However, the uncertainty is not eliminated! Simulations help us to make the lost information visible, such as the range the results may take or the correlations between the model variables. In no case, simulations increase the complexity of the system, nor can they reduce it.

 

Be attentive if you encounter concepts such as "complexity reduction", which are based on the non-use of certain methods. What is claimed as a reduction is basically a loss of information. Encounter the complexity openly and learn to deal with it better. Simulations, according to our conviction, enable you to do so.

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