Simulation is the best way to allocate trucks to shovels in open-pit mine operations

1. The problem

Efficient open-pit mine operations maintain a steady ore feed to the extraction plant (the �bottleneck� as stated in one of my previous postings in this BLOG). This can be guaranteed by allocating sufficient resources (trucks and shovels) to the appropriate circuits. Since hauling represents 50% or more of the total operating costs, economic penalties are incurred when extra resources are assigned. Therefore, an important operational objective is to feed the plant with minimum resources. 

To do this, a two-stage problem is usually formulated:

1. Allocation: Trucks are assigned to shovels according to performance variables of the shovel, desired production levels, and truck cycle times. Successful truck allocation can have a significant impact on the overall performance of the mine. The allocation process is based on historical information and is performed usually at the beginning of each shift.
2. Dispatching: Is a real-time decision making process that dynamically allocates trucks in response to unexpected events or changes in the planned scheme.

Cycle times play an important role in truck allocation and refers to the sum of: travel time from the dumping point to shovel, waiting time at the shovel, loading time, and travel time from the shovel to the dumping point. Due to the inherent variability of mining operations, cycle times are stochastic in nature. To mitigate the risk of not meeting the ore demand due this variability, mine managers often assign extra trucks to haul ore material. This inefficient approach generates long truck queues throughout the mine. As well, when fewer trucks are available for hauling waste material the overall mine planning is negatively affected.

So, the allocation problem can be stated as: �Given a number of available trucks at the beginning of the shift, how to allocate them to have a steady ore feed to the processing plant and to maximize the waste removal.�

2. The usual solutions

Typically, dispatchers allocate trucks at the beginning of the shifts based on historical data and experience. This heuristic approach is inefficient since it relies on the dispatchers' experience, which varies among shifts. On the other hand, many authors affirm that initial truck allocation can be improved by using mathematical programming. There are two main methods proposed: deterministic and stochastic.

Multiple problems arise when using these approaches:

• It is quite difficult (in my opinion almost impossible) to state an accurate mathematical model to represent mine operations.
• Most of the assumptions made to build these models are unrealistic and tend to oversimplify the reality.
• Truck allocation can only be implemented using a discrete number of trucks, and therefore fractional results are not acceptable. Approximations of these fractional results can make the solution non-optimal or unfeasible. As well, integer programming can make problems computationally intractable.
• Deterministic approaches do not include the inherent variability of mining processes (average cycle time is used to state deterministic problems). Random changes due to variability can make the deterministic optimal solution non-optimal (and in some cases unfeasible!)
• It is not always clear if working with stochastic programming will provide an appreciable benefit as a worthwhile trade-off for its complexity. Stochastic programming typically demands heavy computer resources, particularly if there are many realizations to be evaluated.
• In both cases, there is no assurance that the model will converge to the optimum solution within a reasonable number of iterations.

3. The usual suspects: normal and exponential distributions

Typically, stochastic truck allocation models are formulated based on the following assumptions in regard with the variables (e.g. truckloads, cycle times, loading times):

• are normally distributed (models based on �queuing theory� assume that service times are exponentially distributed),
• have known standard deviation, and
• vary independently from each other.

These assumptions make models simpler to be stated and computationally tractable. Most textbooks and papers present these neat formulations without questioning the validity of the assumptions.

The problem is that these distributions (i.e. normal and exponential) have no relation at all with what happens in the reality! Indeed, the range of a normal distribution is from positive to negative infinity. To my understanding there are not negative times . . . are they?

As well, the assumption that all variables are independent it is not true since there is always a certain degree of correlation among them.

Finally, there is no reason to expect that real-world stochastic processes vary in accordance to some theoretical distribution. Those who are familiarized with distribution fitting know that sometimes a triangular approximation is needed, or a empirical distribution is the best alternative. There is no way to mathematically model these situations without doing the above detailed unrealistic assumptions.

4. Computer Simulation: the right tool to decide truck allocation

Is there a way to decide truck allocation in a more accurate and efficient way? The answer is YES!

Computer Simulation has been widely used to produce experimental data for validating and evaluating different operating policies and dispatching algorithms without interfering in the real mine operations. As well, computer simulation can be used for a careful evaluation of possible combination of shovels and trucks in order to achieve minimum production costs and reduce capital expenditures.

Depending on the completeness of the simulation model, mine operations can be digitally imitated with great accuracy without the need of making unrealistic assumptions. One of the most important aspects of simulation, is the reliability of the results produced! In particular, let suppose that we have a comprehensive simulation model that imitates the real mine operations in a highly accurate way. Let�s assume as well that the model has been properly validated and calibrated and it is ready to use. Can we use this model to find the best truck allocation? Again, the answer is YES.

The simulation model can be used to test different allocation schemes and find the most appropriate one, considering the whole variability inherent to mine operations (e.g. loading times, equipment down time, travel times).

HERE you can find an example of a standardized and highly efficient model that can be adapted to any mid-size mine. This model has a built-in �optimizer� that performs an �intelligent search� among different allocation configurations to find the best one.

In conclusion, truck allocation plays an important role in reducing the operational costs in open-pit mine operations and maintaining a steady ore feed to the extraction plant. Different ways to perform truck allocation can be used ranging from manual allocation (using the dispatcher�s experience) to allocation assisted by complex mathematical programming. Usually these mathematical models are stated based on unrealistic assumptions that prevent them to achieve the right results. Computer simulation can help us in this endeavour by accurately imitating the real operations including all the variability. Finally, the good news is that SmartSimulation has a very comprehensive simulation model that can accurately find the best truck allocation!

Rene Alvarez, IE, MEng
www.SmartSimulation.ca