Optimization-based dispatching policies for open-pit mining

Abstract

We propose, implement, and test two approaches for dispatching trucks in an open-pit mining operation. The first approach relies on a nonlinear optimization model that incorporates queueing effects to set target average flow rates between mine locations. The second approach is based on a time-discretized mixed integer programming (MIP) model. The MIP model is difficult to solve exactly in real-time operations, so we present a heuristic that quickly produces high-quality feasible solutions. We test the dispatching policies by building a discrete-event simulation model of an open-pit mine. We present a full computational study of the two policies in which we perform output analysis on key metrics of the open-pit mine simulation. Results indicate that the MIP-based dispatching policy consistently outperforms the more commonly-used average flow rate dispatching policy on open-pit mines in a wide variety of operations settings.

Appendices

Appendix A: Additional computational experiments

1.1 A.1: Heuristic solution quality

In this appendix, we report on an experiment that demonstrates the performance of the RINS heuristic from Sect. 3.4, and we use the solutions to measure and assess the quality of our MIP model’s approximation of the nonconvex blending condition (52).

In our method, we allow the production rate target objective (MBLP-1) to solve to optimality and enforce a 20-s time limit on the extraction targets (MBLP-2). If a solution is not found in the 20-s time limit, we allow Gurobi to continue searching for a feasible solution for up to 90 s. If $t'$ is the number of seconds required to solve (MBLP-2), then we set a time limit of $t^{\max }=90-t'$ s for solving (MBLP-3). Thus, a time limit of 90 s is given to solving both (MBLP-2) and (MBLP-3). In all our experience, the problem (MBLP-1) solves in a matter of seconds. We evaluate two heuristics for solving (MBLP-3):

• 1. Use Gurobi to directly solve (MBLP-3) with a time limit of $t^{\max }$ with the parameter MIPFocus=1 to emphasize finding feasible solutions quickly;

• 2. Use the RINS heuristic described in Sect. 3.4 to solve (MBLP-3) with a time limit of $t^{\max }$ and with MIPFocus=1.

With the exception of the MIPFocus parameter, all Gurobi parameters are left in their default state.

The mine configuration used in this experiment is the same for all of our computational experiments, with $|G| = 5$, $|B| = 5$, $|P| = 2$, $|S| = 2$, $|D| = 5$, $|R| = 1$, and $|K|=2$. We test planning horizon lengths of 20 min and 40 min using 30-s time periods, so $T^{\max } \in \{40,80\}$. We generate 10 separate instances of the random input parameters for each planning horizon length, for a total of 30 distinct instances. For the ore quality constraints (Eqs. 55–57), we choose the periods covered by each constraint (H) to be 20 periods (10 min) and the distance between successive constraints (F) to be 4 periods (2 min). For more information on these choices, the interested reader is directed to the thesis (Smith 2019), which contains a wealth of experiments aimed at tuning model and algorithm parameters.

For each planning horizon length, we observe the total quality target violation in the solution to (MBLP-2), in the solution to (MBLP-3) when solved as close to optimality as possible in 1 h (OPT), in the solution to (MBLP-3) using a time limit of $t^{\max }$ (90-SEC), and in the solution to (MBLP-3) using a time limit of $t^{\max }$ with the RINS heuristic (RINS). The results of this experiment can be found in Table 6.

Table 6 Summary of the total quality target violation objective using different solution methods compared to not solving (MBLP-3)

From the results of the experiment, we observe that the ore quality objective improves significantly when solving (MBLP-3) rather than stopping after solving (MBLP-2). In many cases, the quality of the solution found by Gurobi in 90 s and the RINS heuristic in 90 s are quite comparable and not substantially worse than the solution found after running Gurobi for 1 h. However, when using the 90-SEC heuristic, no feasible solution was found in five of the 40-min horizon instances. We were able to find a solution within 90 s for all instances using the RINS heuristic. Thus, we use the RINS heuristic with a time limit of 90 s for all remaining computations.

Recall that the quality violation measurements in (MBLP-3) are only an approximation of the true ore quality because of the nonconvex nature of the blending model. Because this approximation does not accurately calculate the true quality violations, after solving (MBLP-3) we use Eq. (52) to calculate the true ore quality in the given solution. We demonstrate the solution quality of each instance by reporting for each instance the percentage of the time horizon in which the true ore quality at the processing sites was within 1% of the target. Recall that we set the window value $\epsilon = 0.5\%$ in Eqs. (53)—(56), so this is twice the window size in which the model attempts to keep the quality. The results of this calculation are summarized in Table 7.

Table 7 Total percentage of time the ore-quality was within 1% of the ore-quality targets at the processing sites

The results of the experiment indicate that solutions from the RINS heuristic keep the ore quality close to the target at the processing sites. The fifth instances is paradoxical, since for the solutions to the 40 min horizon problem, the solution without considering ore-quality has a larger fraction of time in which the ore quality in within a 1% band of the target at the processors. In this instance, the solution of the (MBLP-2) problem had significant violations at the end of the horizon, but the violation was small a large fraction of the time.

The results presented in this section allow us to draw some conclusions about our model. As expected, the (MBLP-3) solution quality deteriorates somewhat when solved with the RINS heuristic, but overall the solutions are comparable to solving the instance as close to optimality as possible with a 1-h time limit. More significantly, these solutions are obtained in 90 s or less, making this model efficiently implementable. For the horizon of 40 min, the solution obtained from RINS within 90 s is better than the solution obtained by simply enforcing a 90-s on the emphasize feasibility method, since for many instances a solution could not even be found in that time. Furthermore, we conclude that the inclusion of (MBLP-3) is important for meeting the ore quality objective, in spite of the additional computational difficulty over (MBLP-1) and (MBLP-2). Results also demonstrate that the moving and overlapping window approximation to the nonconvex blending constraints ensure tight adherence to ore quality targets at the processors.

1.2 A. 2: Evaluation of model components

Using the simulation, we also evaluated the impact of novel features of the nonlinear average flow-rate model and the discrete-time MIP dispatching model. Specifically, we test the effect of the queueing constraints (9)–(12) in the nonlinear flow-rate model and the moving window truck capacity constraints (39)–(46) and ore quality constraints (53)–(56) in the MIP dispatching model. We evaluate each policy on the same instance described in Sect. 4.3. In these tests, we consider cases where the mine has 15, 25, or 35 trucks. In all cases, we run 30 replications of the simulation to estimate the mean value of the six key metrics, and we plot confidence intervals for these estimates in three panels, representing the evaluation of the instance for 15, 25, and 35, trucks, respectively.

1.2.1 A.2.1: Queuing constraints average flow-rate matching policy

Our first experiment demonstrates the utility of the queueing constraints in the nonlinear average flow rate model for the two-phase policy of Sect. 2.1. We set the target rates for AFRM by considering the effects of queueing (Q), by including constraints (9)–(12), and without these constraints (NQ) . Figure 13 displays the shutdown and slowdown metrics for this study.

Fig. 13

Comparing shutdown/slowdown metrics for nonlinear flow-rate model target-matching policy on processing metrics with queuing constraints (Q) and without queuing constraints (NQ). Three panels are for 15, 25, and 35 trucks, respectively

We observe that the queueing constraints make no difference in our ability to meet processing targets for 25 and 35 trucks. When only 15 trucks are available, however, use of queueing constraints significantly improves the efficiency of the system. With the queueing constraints, neither of the processing sites is ever shut down and the percentage of time required to slow down reduced from nearly 50–30%. The mining site target extraction amount and ore quality metrics are similar with and without the queueing constraints added. This experiment clearly demonstrates the positive impact of specifically considering queueing considerations in the average flow-rate model.

1.2.2 A.2.2: Capacity constraints in the MIP dispatching model

To evaluate the effectiveness of the moving window capacity constraints, we run the simulation first by simply rounding up the truck capacity in each time period at each location to the nearest integer (Round). We then run the simulation with moving window constraints (constraints 39–46) that attempt to limit the “round-up error” (MW). Figures 14, 15, and 16 display the results of this study.

Fig. 14

Comparing performance of discrete-time MIP policy on processing metrics with moving window capacity constraints (MW) and with only per-period capacity constraints (Round). Panels are for 15, 25, and 35 trucks, respectively

Fig. 15

Comparing performance of discrete-time MIP policy on mining target metrics with moving window capacity constraints (MW) and with only per-period capacity constraints (Round). Panels are for 15, 25, and 35 trucks, respectively

Fig. 16

Comparing performance of discrete-time MIP policy on ore-quality metrics with moving window capacity constraints (MW) and with only per-period capacity constraints (Round). Panels are for 15, 25, and 35 trucks, respectively

In Fig. 14, we observe that the capacity constraints make no difference in our ability to meet processing targets for 25 and 35 trucks. When only 15 trucks are available, however, use of capacity constraints significantly improves the system processing performance. In Figs. 15 and 16, we see that the two methods are nearly indistinguishable with respect to the mining site extraction and ore quality metrics, although the MW variation does appear to over-mine slightly more than simply rounding up. We conclude that the moving window capacity constraints are critical to meeting the processing criteria when few trucks are available, but the use of the constraints has less impact when there sufficiently many trucks.

1.2.3 A.2.3: Ore quality constraints in the MIP dispatching model

To evaluate the effectiveness of the moving window ore quality constraints (constraints 53–57), we run the simulation first by stopping after minimizing the mining target deviation (MBLP-2) rather than continuing to minimize the ore quality target violation (MBLP-3). This case is labeled (None) in the result panels. We then run the simulation using the moving window constraints that approximate ore quality violation (MW). Implementing the ore-quality minimization phase has no impact on the system’s ability to meet processing targets, as neither processing site is slowed down nor shut down in any of the 30 simulation replicates. Figures 17 and 18 display the confidence intervals of estimates of the mining site target and ore quality metrics.

Fig. 17

Comparing performance of discrete-time MIP policy on mining target metrics with moving window ore quality constraints (MW) and without the constraints (None). Panels are for 15, 25, and 35 trucks, respectively

Fig. 18

Comparing performance of discrete-time MIP policy on ore-quality metrics with moving window ore quality constraints (MW) and without the constraints (None). Panels are for 15, 25, and 35 trucks, respectively

Interestingly, although we would expect the MW variation to perform slightly worse on the mining metrics, in Fig. 17, we see that the two variations are nearly indistinguishable in terms of under-mining, and the MW constraints in the simulation actually seem to reduce the over-mining metric. As expected, Fig. 18 demonstrates that the ore quality values significantly improve when using the moving window ore quality constraints regardless of the number of trucks. We conclude that the moving window ore quality constraints are critical to meeting the ore quality targets and, surprisingly, might even improve the mining extraction target metrics.

Appendix B: Instance data

This appendix describes how we created families of instances for testing the models and dispatching methods. In Table 8, we give each instance parameter along with a brief description of how the parameter is used. We also give the range of values this parameter can take and the method we use to generate the parameter. If the parameter generation method is “Fx”, then the parameter value is given in the Range column. If the parameter generation methods is “UC,” then the parameters are randomly generated from a continuous uniform distribution with the bounds given in the Range column. “UI” parameters are randomly generated from an integer uniform distribution with the bounds given in the “Range” column. The total number of trucks at different locations are generated as specified in the table, and their specific locations are assigned at random. To generate mining extraction targets, each mining site is given a “size” parameter, $s_i$, and its relative size $\psi _i = s_i/\sum _{i} s_i$ is used in the extraction target calculation. Mining locations in the mine are randomly generated on a circular map between 0.9 and 1.5 miles from the origin. Processing, stockpile, and dump sites are randomly generated on the map between 0.2 and 1.2 miles from the origin. The distance is calculated as the (fixed) Euclidean distance between two locations. The conversion factor of 4/3 in the translation to periods $\theta _{ij}$, assumes that trucks travel at 40mph.

Table 8 Parameter ranges for instances used in computational experiments