# Estimating the Optimal Overall Slope Angle of Open-Pit Mines with Probabilistic Analysis
## 5 Conclusions  
This paper proposes a methodology for evaluating open-pit mine stability at various overall slope angles in terms of safety, productivity, and cost. Due to the heterogeneity of the rock mass, a hybrid approach method was used, in which deterministic numerical modelling was combined with probabilistic methods. At various slope angles, three failure evaluation factors were investigated: factor of safety (**FOS**), maximum shear strain and incremental displacements. The slope failure probability was calculated using a minimum FoS of 1.5. When overall slope angles range from 20\({}^{\circ}\) to 35\({}^{\circ}\), the results show that the probability of slope failure, Pf, is rare to unlikely (e.g., 57\(\%\) \(<\) Pf \(<\) 20\(\%\)). As the slope angle increases (e.g., from 40\({}^{\circ}\) to 55\({}^{\circ}\)), the probability of slope failure, Pf, increases (e.g., 20\(\%\) \(<\) Pf \(<\) 60\(\%\)). At steep slope angles, the probability of slope failure, Pf, increases (e.g., 60\(\%\) \(<\) Pf \(<\) 85\(\%\)) (e.g., 60\({}^{\circ}\) and 65\({}^{\circ}\)). The probability of slope failure, Pf, is certain at very steep slope angles (e.g., 70\({}^{\circ}\)) (e.g., Pf \(>\) 85\(\%\)). For safety purposes, designing open pits at steep to very steep angles (e.g., \(\geq\)60\({}^{\circ}\)) is not recommended. In terms of productivity and profitability, the results show that mine productivity increases and the cost of overburden removal decreases as the slope angle increases. However, slope stability suffers. For instance, at the steepest slope angle (e.g., 70\({}^{\circ}\)) the mine net profitability, cost of overburden removal and probability of instability, Pf, are $2,228,800, $120,200 and 99.04\(\%\), compared to $1,441,400, $907,600 and 0.89\(\%\) at a slope angle of 20\({}^{\circ}\), respectively.  
The proposed method was replicated using a real case study open-pit mine. The effect of groundwater and seismicity was examined. The results show that there are multiple unstable slip surfaces. Because of the influence of groundwater, the minimum **FOS** is 1.265. Seismicity demonstrates that **FOS** decreases to 0.154. (e.g., critical seismic coefficient). Deterministic analysis reveals multiple slip surfaces, two of which fail. Probabilistic  
Figure 18: Histogram of the FoS from the 3D probabilistic analysis.  
analysis, on the other hand, produces only one failed slip surface. Uncertainty associated with rock mass reduces **FOS** from 1.414 (deterministic) to 1.246 (probabilistic), with a failure probability of 16%. In terms of cost and profitability, the total estimate of overburden removal and ore recovery are $23.5 \(\times\) 10\({}^{6}\) and $320 \(\times\) 10\({}^{6}\), respectively. Consequently, the net profit will be $296.5 \(\times\) 10\({}^{6}\).  
Conceptualization, W.R.A., C.H. and M.A.M.A.; methodology, W.R.A., C.H. and M.A.M.A.; investigation, W.R.A., A.S., A.R.T. and M.A.M.A.; writing--original draft preparation, W.R.A., A.R.T., A.S. and M.A.M.A.; writing--review and editing, W.R.A., A.R.T. and C.H.; visualization, W.R.A., M.A.M.A., A.S. and A.R.T.; supervision, W.R.A., A.R.T., C.H. and M.A.M.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.