The Monte Carlo Method

Monte Carlo analysis is a technique where a model is created that simulates a particular process or business case, and the model is then run a large number of times to generate an expected output value. The inputs to the model are generated at random based on an understanding of the values that they may actually take e.g. only within an allowable range, or clustered around a typical expected value. By running the model a large number of times, we can get an appreciation of the variability of the output value based on the variability in the input values. This technique enables us to generate an answer which is very close to reality in a fraction of the time and cost it would take to perform a full analysis.

In industry, it is often not possible to model every single scenario that could possibly arise because there are just too many possibilities to consider. Consider a 5 step process where each step can take one of 20 different values. The total number of combinations therefore is 205, which is 3.2 million possible different outcomes. Even if a computer could run a full simulation every second, it would still take 37 days at 100% utilisation to calculate in full. In business, we need to be able to make decisions in a shorter timeframe than this.

The Monte Carlo technique can be applied to almost any scenario. There is uncertainty in almost every aspect of business (and life in general), whether it's logistics, designing products or forecasting profits etc etc. In any situation where we need to understand the effects of uncertainty, Monte Carlo analysis is a great tool to have in the arsenal. There is even a saying that goes "All roads lead to Monte Carlo" because it is so simple, so versatile and so useful.

Example - Business Simulation

Suppose you run a business that manufactures metal beams that are 3 metres long from three separate beams provided by suppliers. The finished beams are sold for £100 each, and must be within +/- 1mm tolerance, or else your customers will refuse to buy them.

You have a choice of three suppliers to provide your input beams. They can all manufacture their beams to the length required, but with different levels of accuracy*. The details for each supplier are listed below:

*The accuracy of each supplier is quoted as the standard deviation of their finished product.

Supplier Cost Accuracy
Cheap £20 +/- 1.0 mm
Medium £25 +/- 0.5 mm
Expensive £33 +/- 0.1 mm

So - Can we balance the probability of being outside of the required tolerance against the costs of the different suppliers? And can we then optimise the system to maximise our profit margin? Of course we can - using the Monte Carlo analysis technique.

Quick note: In every scenario given below, the analysis has been modelled based on the "manufacture" of 1,000 beams. Each chart shows the distribution of the three input beams on the top row, and the distribution in final output on the bottom. The dashed black lines are the limits above/below which our customers will not buy our product.

All Cheap Beams

On smaller devices, best to open chart directly.

If we buy all of our beams from the cheapest supplier with the lowest accuracy, then we risk our final product being outside of the required specification, unable to be sold. Modelling this situation, it turns out that only around 45.5% will be accpetable for sale, so we're generating less than half of the total revenue possible.

All Expensive Beams

On smaller devices, best to open chart directly.

If we buy all of our beams from the most expensive supplier, then we should be able to sell all of our manufactured products, but they end up being so expensive to manufacture that we don't make much money. This is still preferable to the previous scenario, but surely we can still do better.

All Medium Beams

On smaller devices, best to open chart directly.

If we buy all of our beams from the medium supplier, then we should have a good balance of quality products and reasonable manufacturing costs, right? Wrong. Turns out that the final output is still unacceptably spread-out, leading to reduced acceptance rates and therefore reduced profits.

Best Solution

On smaller devices, best to open chart directly.

Running the analysis for every possible combination of these suppliers, it turns out that the best combination is one beam from the middle supplier and two beams from the expensive supplier. This leads to a profit margin of 2.8%, which is more than 2.5 times greater than the previous best estimate.

The Optimum Solution - Design of Tolerances

Looking at the suppliers, it is clear to see that the tighter their accuracy, the more expensive their product is. We know what our required output tolerance is - that is, the level of variation that is acceptable to our customers (+/- 1mm). Therefore, can we work out what exact level of variation we can tolerate in our inputs to optimise our business case?

Fitting a curve through the cost vs tolerance points for these three suppliers, we can generate a continuous relationship that can be used to trade the two quantities against each other. Therefore, we can design the cheapest possible part that will still meet our requirements, thus maximising our profits.

Running a sweep through the tolerance range from 0.1mm to 0.5mm, we can estimate the expected returns for each level of tolerance for both cost and acceptance rate. Every scenario was modelled with the manufacture of 1,000 beams, and each scenario was run 100 times to give a feel for the variation within each prediction.

On smaller devices, best to open chart directly.

The Optimum Solution

On smaller devices, best to open chart directly.

This analysis shows us that the optimum tolerance is around 0.28mm, which generates expected returns of a massive 11.5% - that's over 4 times greater than the previous best solution!

Furthermore - since none of our suppliers currently offer this tolerance value, we can engage with them to negotiate a price for such a product, and if we can create some competition between suppliers, that can drive the price further down, further improving our profit margin.

Conclusion

By modelling our business and running each of the viable scenarios a finite but robust number of times, we have ensured that our business is profitable. Furthermore, we have been able to identify an opportunity by which we can improve our profit margins significantly above anything that was possible when we first entered business. Not bad for one evening's work...