Drawing Better Lines in the Pit: Optimizing Dig Limits When Ore Has Many Destinations

Every day on an open pit bench, someone decides where each piece of blasted rock goes. Some goes to the mill, some to a leach pad, some to a low-grade stockpile, and the rest to the waste dump. Those decisions look small, but they add up. Send ore to the waste dump and the value is gone for good. Send waste to the mill and you pay to process rock that returns nothing. In my master's thesis with my advisor, Dr. Kwame Awuah-Offei, at Missouri University of Science and Technology, we asked a practical question: how can we draw these boundaries optimally when ore can go to many different destinations and the digging equipment has real size limits? What Are Dig Limits? Mines model each bench as a grid of small blocks, and each block has an estimated grade. If you classified each block on its own, you'd get a patchwork map where every block is labeled as mill ore, leach material, or waste. The problem is that an excavator can't dig a single small block and send it somewhere different from its neighbors. Its bucket and reach mean it has to take material in groups. So blocks must be combined into dig limits: polygons that are large enough for the equipment to mine as a unit, with every block inside sent to the same destination. Traditionally, geologists draw these boundaries by hand using blast hole samples. That takes a lot of skill, but it is slow and subjective. Hand-drawn limits also rarely find the most profitable arrangement. The Gap We Wanted to Fill Researchers have tried to automate dig limits with simulated annealing, genetic algorithms, heuristics, and clustering. These methods are useful, but none of them guarantees the best possible answer. The one earlier approach that does guarantee optimality, a mixed-integer linear programming model by Sari and Kumral (2018), has two limitations: It only handles two categories, ore and waste. Real mines are more complicated. The copper deposit in our study has seven possible destinations: mill ore, sulfide crushed leach, oxide crushed leach, sulfide run-of-mine leach, oxide run-of-mine leach, low-grade stockpile, and waste dump. It only allows square mining areas. It also uses variables with four indices, so the model grows quickly as the bench gets larger. Our Approach: An Idea Borrowed From Scheduling We formulated the problem as a binary integer linear program (BILP). For each block and each possible destination, the model makes a yes-or-no decision. The most important part of any dig limits model is the minimum mining width: the rule that every mined area must be at least as large as the equipment needs. Writing this rule efficiently is difficult. Direct formulations can require an exponential number of constraints, and solving becomes impractical. Our solution came from an unexpected field: production scheduling. Queyranne and Wolsey developed compact formulations for machines that, once switched on, must stay on for a minimum length of time. A run of blocks along a row is mathematically similar to a machine's on-time: once the model "starts" mining a stretch of blocks for a destination, that stretch must continue for at least the minimum width. We adapted this idea from one dimension to two. The model tracks where each mined area starts along both directions of the bench, so the minimum width is enforced as a rectangle, which doesn't have to be a square. For example, a 2×5 block rectangle can suit equipment that works better along one axis. Putting It to the Test We tested the model on bench sections from a real porphyry copper-molybdenum block model. We implemented it in MATLAB and solved it with the Gurobi optimizer. Because the mine's actual processing recoveries are confidential, we assigned illustrative economic values that match the mine's own "best classification" of each block. The waste dump was left out as a selective destination, since waste does not need to be carefully dug. Anything not assigned to a paying destination can simply be removed afterward. For a 20×20 bench section with a 3×3 minimum mining width, the model found the proven optimal solution in about 64 seconds, with a 0.0% optimality gap. The way it reached that solution is worth a look. Where small patches of higher-value material were too narrow to mine on their own, the model reclassified them to match their surroundings. The result gives up a little value on paper, but every boundary it draws can be dug by the equipment. In other areas, it sent blocks to their second-best destination when that allowed a larger, minable area that was more profitable overall. This is the kind of trade-off a skilled planner makes by instinct. The model makes it systematically and proves the answer is the best one available. What We Learned 1. Larger benches take longer to solve. Going from a 20×20 section (400 blocks) to 30×30 (900 blocks) raised the solve time from about 1 minute to about 4 minutes. Four minutes is workable for short-term planning, but real benches can be much larger. Faster solving is a priority for future work. 2. Larger minimum widths reduce value. As the minimum mining width grew from 3×3 to 5×5, the bench's value dropped in every case, because larger dig limits mean less selectivity. The practical lesson for planners is to use the smallest mining width the equipment can realistically handle, not a conservative rounded-up number. 3. Aligning the rectangle with the geology adds value. This was the most interesting result. On the 30×30 section, a 2×5 minimum width produced about 2.6% more value than a 3×3 square. The same rectangle turned 90 degrees (5×2) produced about 4% less value than the 2×5 orientation. The reason is the geology: in this deposit, similar material tends to run in one direction, and a mining shape aligned with that trend captures it more cleanly. This suggests that models allowing only square mining areas may be leaving money in the ground whenever the equipment could work with a rectangular footprint. Limitations and Next Steps No model captures everything. Ours does not account for the possibility that narrow, elongated dig limits slow down excavation and raise mining costs. It also treats block values as known, when in reality they are estimates with uncertainty. And it doesn't yet account for how rock moves during blasting, which can shift material across the boundaries we draw. Our next steps are to reduce solve times for larger benches, add blast movement, and incorporate grade uncertainty and risk. Why This Matters Dig limits sit where planning meets daily operations. They are the point where a block model becomes actual truckloads. As mining grows in new regions, including here in Saudi Arabia, tools that turn geological knowledge into optimal, equipment-feasible decisions will be an important part of running mines that are both profitable and responsible with their resources. The MATLAB code is openly available on GitHub for anyone who wants to try it on their own data. Questions, or interested in collaborating on mine planning optimization? Get in touch.