Theory Behind

1. The theory behind the technology

Now you have practiced for a while, it is time to get deep into the theory behind the MiningMath technology.

1.1. MiningMath Uniqueness

MiningMath optimization is not constrained by arbitrary decisions for cut-off grades or pushbacks, since these decisions are usually guided by prior knowledge or automated trial-and-error. Thus, each set of constraints in our technology has the potential to deliver an entirely new project development, including economic, technical, and socio-environmental indicators, along with a mine schedule, while aiming to maximize the project's NPV.

License to operate is the main risk for the mining industry in 2019-20, according to EY. Any mining company willing to become a market leader should be committed to incorporating socio-environmental aspects into a strategic evaluation, quantifying possibilities, and impacts to discuss with society. This is only possible by bringing mathematical optimization into the decision-making process.

1.2. What is Direct Block Scheduling?

During decades, the mining industry has dealt with Mine Planning as a step-by-step process. This traditional technology has been intelligently established in the face of the technological limitations of that time. The conventional methodology, portrayed in Figure 1, in general, consists of three main stages: (1) pit optimization with nested pits (using Lerchs-Grossman (LG) algorithm); (2) definition of pushbacks; and (3) scheduling of benches within pushbacks. Intermediate manipulations and cycles over these steps might be required to achieve a higher NPV.

Figure 1: Traditional scheduling process.

Direct Block Scheduling (DBS) is an alternative approach to this step-wise process. It has been studied for almost 50 years by researchers worldwide, but back then computers were not developed enough to handle the proposal first made by Johnson in 1968, of a global mathematical model to optimize mining projects. Over the decades, other authors followed Johnson's approach and introduced their algorithms. Technology advanced as well, but solving larger problems remained a challenge.

DBS became feasible only after the advent of 64-bits technology and, in 2015, MiningMath DBS, implemented with DBS technology, was officially released into the market. The MiningMath technology, coupled with the DBS, considers all periods simultaneously, providing a holistic view of the mine scheduling problem by maximizing the NPV, unconstrained to predefined pushbacks and cut-offs. Figure 2 summarizes DBS's studies over the last decades:

Figure 2: Research on Direct Block Scheduling over time.
Figure 3: Comparison between traditional and a DBS methodology applied to the whole mining problem. Source: Morales et al., 2015.

MiningMath's goal is to define the pit limit and mine schedule simultaneously, that is, to determine which blocks should be mined, when this should happen and to where they should be sent to maximize the NPV, while respecting production and operational constraints, slope angles, discount rate, stockpiles, among others, all performed straight from the block model. This means that the steps of pit optimization, pushback and scheduling are not obtained separately, but in a single and optimized process.

In addition, the algorithm's framework, based on mixed-integer linear programming (MILP) with heuristics, is flexible to include any sort of other constraints (fleet and excavation hours, metal production, average haul distance, among others) and blending. Figure 3 illustrates a comparison between DBS and traditional methodology.

Direct Block Scheduling does not require predefining your destinations, since it is capable of automatically performing the ore/waste delineation. Because of this optimized definition, the economic values are calculated for each possibility before importing the data. This means that N different destinations can be created, leaving for the algorithm the duty to define the best blocks’ destinations based on the feasibility to mine them and their economic contributions, represented by the block value. The user no longer needs to assume a certain cut off-grade based on previous experience to predefine whether a block is an ore or waste.

The average grade reported on each period by MiningMath (Figure 4) can be interpreted as an “optimal” cut-off that was achieved as a consequence of a complex optimization process that considered production, geotechnical and temporal constraints. They oppose the assumed fixed arbitrary value that would govern the blocks' destinations, present in the conventional methodology for mine scheduling. Figure 5 illustrates a simplified flowchart of how blocks' destinations are defined.

Figure 4: Average grade throughout the mine life achieved by MiningMath on McLaughlin deposit.
Figure 5: Simplified flow-chart of blocks' destinations optimization.

1.3. Discounted Cash flow x Undiscounted Cash flow

The use of LG/Pseudoflow methods to perform pit optimization aims to maximize the undiscounted cash flow of the project. On the other hand, MiningMath maximizes the discounted cash flow, including all the constraints that you might be considering in your case. Therefore, regions in which MiningMath has decided not to mine are, probably, regions where you have to pay for removing waste on the earlier periods, but the profit obtained by the discounted revenue from the hidden ore does not pay for the extraction.

A proper comparison between this methodology could be done if you import the final pit surface obtained from the other mining package into MiningMath, and use it as Force/Restrict mining. This way, MiningMath will do the schedule optimization using the exact same surface, which will allow you to compare the NPV for each case.

Figure 6: Undiscounted versus discounted cash flow optimization.
Figure 7: Undiscounted versus discounted cash flow optimization regarding a minimum mining width.

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    • MORALES, N.; JÉLVEZ, E.; NANCEL, P.; MARINHO, A.; GUIMARÃES, O. A, (2015). A Comparison of Conventional and Direct Block Scheduling Methods for Open Pit Mine Production Scheduling. Proc. APCOM 2015, Alaska.