THE MOST DIFFICULT TASK TO DEAL WITH WHEN SOLVING COMPLEX PROBLEMS

INTRODUCTION
Complex systems are neat nonsense systems that have a number of elements or properties that are often true for such systems – namely nonlinearity, chaotic behaviour, many feedback loops, power-law (or at least, scale-free) behaviour, and network effects – that interact to produce the behaviour of the whole over time. Based on this we can define complex dynamic systems as complex systems whose behaviour evolves over time.

Complex problems are questions or issues that cannot be answered through simple logical procedures. They generally require abstract reasoning to be applied through multiple frames of reference.

An example of a complex dynamic system/process is the mine supply/demand chain (see Figure 1). This is because the different stages necessary to find, extract, process, and sell the final product in the market, produce different sources of data that will vary over time.

Figure 1. Mining supply/demand chain.

The mine project evaluation (i.e., mine plan, design and valuation) – see Figure 2 for a model – is an example of a complex problem, which cannot be solved through simple logical and linear procedures – let alone as a practical process.

Figure 2. A mathematical model to estimate a mining project using real options.

THE DIFFICULT PART OF SOLVING A COMPLEX PROBLEM
Conversely with the traditional way of complex problem solving, I believe the most difficult part of solving a complex problem, is not to find a complex process solution but to develop an easy to follow – user-friendly – solution processes. It will ease the work of decision makers and can be understood by everyone dealing with the problem in practice, i.e., without having the necessary science background.

AN EXAMPLE OF A SIMPLE – USER FRIENDLY – PROCESS THAT SOLVES COMPLEX PROBLEMS
A nice example of a simple and user friendly process, that can be understood by everyone, and that solves complex mathematical problems, without the need of having the mathematical background, is the Pascal’s triangle.

Indeed, the Pascal’s triangle is a special arrangement of numbers (See Figure 3) that can be easily constructed according to the following rule. Start with 1s down both diagonals and then fill in the interior entries by adding the two numbers just above a given position (to the left and to the right). For example the circle number 3 is the result of adding 1 and 2.

Figure 3. Building the Pascal triangle.

Once we got the Pascal’s triangle built to any number of layers, we could use it to solve complex mathematical problems without having the mathematical background, such as the binomial expansion of the form (a+b)^n, for any value of n (see Figure 4). 

Figure 4. Solving the binomial expansion using the Pascal’s triangle.

We could also use the Pascal’s triangle as an excellent way of visualizing the random walk as well as estimating the probability of ending up at a specific location after some steps (see Figure 5). As displayed in Figure 5, The Pascal’s triangle arrangement can used to follow the development of the random walk as the number of steps rises. Each node in the triangle represents a possible ending point of the random walk path, while the values indicates the number of path combinations that can be used to arrive to that location. Our example is showing a possible path (in blue) followed by the red-dot finishing at the location x=-2 after 6 steps, where the value 15 indicates there are 15 path combinations that will lead to arrive to this location, and that the probability for the red-dot to be at this location is 15/64 (where 64 is the sum of all node values of the triangle at the 6th layer or step).

Figure 5. Visualising the random walk using the Pascal triangle. Note the probability for the red-dot to be at its current position (x=-2) after 5 steps is       15  ⁄ 64=0.23.

COMMENTS AND CONCLUSIONS
Mine project evaluation and decision making is critical and non-trivial.
While there is considerable R&D material about the mine evaluation process in the literature realm there is still a lack of an easy to understand and implement  processes and tools. These can allow mine planners and managers not only to solve this complex problem in a simple and practical fashion but also to transmit the benefits of doing this considering all sources of uncertainty – i.e., identifying areas of high risk or high potential in the face of geological, operational and economic uncertainties.

There is no doubt that current efforts are being done to solve the complex problem of evaluating a mining project but not much of this effort (to our knowledge) is addressed to look and develop a simple “Pascal’s triangle” type solution to this complex problem.

It is a technological race where whoever can solve it will for sure provide new frontiers for the future mining industry, a future where price volatility will drive the metal market, and operating cost will increase in order to mine low metal grades at very deep mines, while taking care of the environment.

Anyhow, it will be interesting to see what is coming during the next 5 years.

The Integrated Valuation/Optimisation Framework (IVOF), is a holistic process/tool that considers uncertainty when evaluating mine projects. This is because the IVOF is an advanced data analytic process-engine that uses advanced quantitative simulation-optimisation and real options techniques.

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