ORB. Image: Polymathian
Equipment Technology

Mathematics: the best kept secret in mine optimisation

From digital twins and statistical analysis to short interval control. The team at Polymathian guide us through the world of applied industrial mathematics and how it can transform mining operations

I felt wildly out of my depth when I started researching this feature.

I told my husband this prior to the interview I had arranged with Australian industrial mathematics specialist Polymathian. His response was:

“Carly, you’re smart. But those guys are probably on another level… Like, Sheldon from The Big Bang Theory smart. Just be honest and tell them you know nothing about maths.”

Like me, I expect that 90% of people working in the mining industry have never had reason to learn about applied mathematics, so I eventually decided that honesty was the best policy and, thankfully, the Polymathian team were more than kind to me.

Let me share with you what I learnt…

“Industrial mathematics focuses on problems which come from industry with the objective of developing solutions which are cost efficient, practical and relevant to industry,” Colin Eustace, head of simulation,  started with the basics.

“It involves development of a thorough understanding of the business and technical issues, formulation of complex mathematical models, implementation of deployed software solutions and the communication required to facilitate application of the solution.”

As you will have already gleaned from my choice of theme this month, mathematics, computing and modelling are all intricately linked. The application of mathematics to large and complex industrial problems generally requires a software solution to manage large input data sets, find an optimal solution and allow visualisation, interrogation and execution of the outcomes.

A dashboard from Polymathian's ORB mining optimisation software. Image: Polymathian
An example dashboard from Polymathian’s ORB mining optimisation software. Image: Polymathian

The continual improvement of coverage and quality of operational data for mining operations is an enabler for mathematical models to optimise and enable automation of mining operations.

“None of these (mathematics, modelling, computing) can exist without the others in a real-world context,” Steven Donaldson, partner and co-founder, told me. “Basically, you’re using maths to model a problem and the way you get an answer is using a computer. Mathematics by itself is interesting, but you can’t use a pen and paper to solve meaningfully sized math problems.

“A model is something you would put into computer code; it’s the link between the real world and the mathematics. It brings the computing and the mathematics together to provide value.”

What makes a good mathematician?

Steven explained: “A good industrial mathematician is someone who knows and understands his/her toolkit very well and has the ability to apply the right mathematical technique to the problem they are trying to solve.

“A large part of what we do involves understanding real-world problems; physically going out to a mine site and talking to people.

“When you’re on site, you’re constantly thinking: ‘these are the maths skills and software that I know. How can I use them to model the problem in a way that brings meaningful value to the operation?”

In practice, this requires an understanding of the trade-offs and limitations of both the math and computing, as well how to create a solution in an appropriate time frame with the correct level of detail. It’s no good modelling a problem in a theoretical way, throwing it into a supercomputer and waiting a week for an answer. Although that approach may be fun (for some) it isn’t a practical way to provide value to a client.

Ben Hollis, director and co-founder, explained that experience and education are the two most important factors when selecting mathematical techniques to model a problem and find the optimal solution.

“The problem a customer is tackling could be incredibly varied,” he said. “For example, they might come to you with an optimisation problem. But before you can solve that, you might have to create statistical models of their assets to figure out how they perform, because the customer doesn’t actually know,” he said.

To be able to explain to a customer why their problem is of a particular type and why a particular approach is going to get them the answers they’re seeking requires an added level of skill; not everyone is a born teacher.

Maths, modelling & mining

Traditionally the quality of a mine schedule or plan is dependent on the quality of the engineer and the amount of time available to run various scenarios. Engineers take an iterative approach to creating a schedule, varying inputs after reviewing results in order to manually search for improved solutions, which ultimately cannot be proved to be optimal.

“What’s great about using maths in mining is its ability to give the guaranteed optimal answer,” said Steven. “Traditional planning and scheduling methodologies don’t offer that because they’re based around heuristics instead.

“Sometimes problems are very simple, like digging a decline in the most efficient way. To get real value out of maths, you need a dimension of optimisation; something that can be variable.

“If the problem is overly constrained and there aren’t many variables to select from, there’s not much of a problem to solve. Some humans are very good at solving these simple and moderately complex problems in their heads, and they may find the best solution. But even the most experienced humans still get tired at 2am or distracted when they have to fix a piece of equipment.

“There are some problems that require little more than a simple rule of thumb, but again you can very quickly get to the point where that’s no longer appropriate. A good way to identify when these problems need more advanced optimisation techniques is when you start noticing different staff members do the job at different levels of quality.

That shows there is some complexity there and, if you can get to the point where your system is as good, or nearly as good as the best guy, every time, there’s a lot of value in that.”

Development scheduling using ORB lsoftware. Image: Polymathian
Development scheduling using ORB software. Image: Polymathian

Digital twins: more than machine learning

Digital twins are one of the most widely talked about modelling tools, but they are also one of the most misunderstood, so let’s set the record straight.

“As the term suggests, a digital twin is a digital replica of a physical entity or system,” Colin clarified. “There are many types of information about physical entities that can be digitised and used for different purposes. This can be anything from the physical attributes of an entity, to supplier details, maintenance requirements and history, to a complete record of every aspect of system operations including interactions between system entities.

“As a result, the scope for digital twins, and the use case can be very different for different applications. One way of categorising use cases is to differentiate between digital ‘asset’ twins and digital ‘process’ twins.”

In the mining context, digital asset twins provide a single source for referencing design documentation, supplier details, maintenance requirements and other relevant information, both for fixed plant and mobile equipment.

This enables improvements from collaboration at the design stage to improved visibility of equipment condition. The digital asset twin may even extend to use of historical data to facilitate a predictive maintenance capability.

A digital process twin also includes system operational information for each time the equipment is used and characterisation of unused time. The operational information provides a basis for analysis of past performance, predicting future performance and providing decision support at various levels, from mine site short interval control (SIC) through to strategic business decisions.

The process digital twin can be extended to include decision support functionality, and various mathematical techniques can be applied depending on the type of business analytics required and the system being analysed.

“Statistical analysis is used for descriptive analysis, or to work out what happened, by understanding whether there is a relationship between two data sets and/or the nature of that relationship,” explained Colin. “Once a relationship is established, statistical analysis can also be used to predict future performance (e.g. if system output is found to be steadily increasing by 20 tonnes per day, what is the likely output for tomorrow?)

“Machine learning is used when there is only a partial understanding of the cause and effect relationships in a system. For example: ‘I can’t identify any clear relationships between ball mill input parameters and variable output’.

“Given both system inputs and outputs over a period of time, machine learning can be used to predict the outputs for a new set of inputs.  It can also be used to produce ‘good’ system performance by identifying a set of inputs that produce ‘good’ outputs.”

What are the outcomes if…?

Simulation is used to work out what could happen (or what could have happened), if the cause and effect relationships in the system are understood. Alternative scenario configurations can also be tested to quantify the effects of physical changes to the system or different operational decisions on performance.

“Mathematical optimisation is used to find the set of inputs that delivers the best outcome. Generally, exact optimisation methods that guarantee finding an optimal solution are the method of choice, but approximate or heuristic methods are required for some problems,” added Colin.

“Decision support functionality based on the above mathematical techniques can be used to forecast future performance, identify the effects of equipment and operational changes and optimise operations.”

Digital twins have enormous potential for improving the safety, reliability and profitability of mining operations.

“With sufficient operational data and decision support functionality, process digital twins have the potential to provide a holistic and transparent view of mining operations and make it possible for the best decisions to be made at every step of the process given the information available at the time,” said Colin.

Decision support that can be embedded in a digital twin can provide a range of functionality including:

  • Optimising operational performance; typical deployment for suitable operations create gains in the range of 5-40%
  • Providing forward visibility of short, medium and long-term performance given the current state, planned equipment availability and the range of typical outages, and
  • Linking short interval control (SIC) decisions directly with optimisation of overall business objectives.

Change management, a slow burn

The benefits are all very well, but what about the people?

Mathematics modifies people’s roles and changes what is required of engineers; going forward, we need people who understand how these systems work and look for opportunities to implement them.

Hollis believes that, when it comes to education, the mining industry could be doing a lot more to prepare its workforce for the changes that will inevitably come, from new recruits and practicing engineers all the way to executive teams.

“Change management is as difficult as it’s ever been,” he told me. “Perhaps more so, because some of the digital tools we have now are so sophisticated. The people who have to wield them inside these organisations need a different set of skills. And those people are in high demand.

“We engage with a range of extraordinarily large companies all the way down to really small ones, and there is a vast spectrum of teams inside those companies.

“Some are amazing, and they will drive the product forward faster than we would. Then there are others where management isn’t forceful enough to drive the change forward, or resist for years even though the value is staring them in the face.”

As is so often the case, the companies that do the best in changing are the ones that are exposed to the most commercial pressure. The past year has seen a huge call from investors for greater transparency throughout the mining process at all levels, and Hollis agrees that greater interest in operational and business optimisation can only be a good thing.

“Most of the people we find who make the change difficult are set in their ways,” he said. “As every year goes by that legacy is being slowly eroded.”

A system summary from the BOLT supply chain optimisation software package. Image: Polymathian
A system summary from the BOLT supply chain optimisation software package. Image: Polymathian

Tip of the optimisation iceberg

Colin believes that we are entering a golden age for optimisation in the mining industry where we have sufficient quality data, maturity of systems and industry buy-in to make significant improvements in operational efficiency and business value.

“Most mining operations involve complex operations, interactions and constraints as well as management of equipment and resources to achieve production targets and create value,” he explained. “These operational complexities present an opportunity for industrial mathematics to offer a step change in operational efficiency and ultimately the overall value of a mining operation.

“The potential for the mining industry is a future where the majority of planning and operational decisions are automated or based on decision-support tools that use all of the information available at the time to make the best possible decisions.”

Integration of mathematical techniques that have historically been used in isolation is also becoming more common. An example of this is the use of simulation techniques to provide forward visibility of the range of likely performance for deployed optimisation tools.

Simulation functionality can be used within decision-support tools to seamlessly transition from an actual operations data feed to a range of likely future operations. This provides mine operators with optimised decision support and also forward visibility of possible bounds for future performance, continually aligned with past performance.

Steven is equally optimistic. “Mathematics in the mining industry, particularly in the field of numerical optimisation has an enormous future,” he said. “As computing power continually increases and the power of the mathematical solvers get better, the possibilities are endless.  Perhaps another way to look at this is: when is it not necessary to model a problem with mathematics?”

This is an interesting point; the mining industry has a lot of money to spend and a lot of complex problems to solve. We haven’t been operating optimally for a long time and that’s partly because there’s been no need to focus on best value, until now. We need to get smarter about how we mine.

“The economies of scale in the mining industry mean that, even if you have a specific bespoke problem that’s never been solved before and will never be seen again, there’s probably still value in solving it using mathematical techniques,” said Steven.

With the proliferation of available data, the ever-reducing cost of hardware available in the cloud, and the ever-increasing power of computers, mathematical algorithms can now solve problems in 30 seconds or less which means we are also inching ever closer to real-time optimisation capabilities.

“Our business is certainly being asked to bring optimisation closer to the now. There’s also a lot of value in super-strategic problems which we’re just starting to talk about; mine scheduling for the next 40 years and dispatching equipment so that its always 100% effective. That’s something we couldn’t do before,” concluded Hollis.



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