Chance-Constrained Bi-Objective Evolutionary Optimization for the Open-Pit Mining Operational Planning Problem

Ishara Hewa Pathiranage; Aneta Neumann

cs.NEOct 3, 2026

Chance-Constrained Bi-Objective Evolutionary Optimization for the Open-Pit Mining Operational Planning Problem

Authors: Ishara Hewa Pathiranage, Aneta Neumann

Organizations: Machine Learning and Optimisation, School of Computer Science and Information Technology, Adelaide University, Adelaide, Australia

Abstract

Open-pit mining operational planning involves allocating limited resources while satisfying production, equipment, and ore-quality requirements. Existing approaches often assume deterministic ore grades or rely on simulation-based evaluation under uncertainty. In this paper, we propose a chance-constrained bi-objective formulation of the open-pit mining operational planning problem under uncertain ore grades. We aim to maximize ore production and minimize fleet cost while satisfying stochastic quality requirements through chance constraints. We model ore grades as independent normally distributed random variables and derive a deterministic reformulation of the two-sided chance constraints, avoiding sampling or simulation during fitness evaluation. We evaluate four multi-objective evolutionary algorithms on benchmark instances under different levels and structures of uncertainty. Our results show that NSGA-II and NSGA-III generally obtain the best feasible ore-production values, with NSGA-II requiring less computational time. GSEMO has the lowest computational cost but obtains feasible solutions less consistently and generally provides lower solution quality. We also observe that higher confidence and uncertainty levels reduce feasibility, with the effect depending on the uncertainty structure. The results highlight the importance of considering both the magnitude and structure of uncertainty in short-term open-pit operational planning.

Figures & tables

Instance	# Fronts	# Shovels	Shovel Characteristics	# Attributes	# Trucks	Truck Characteristics
opm1	17	8	(4, 900 t ), (2, 1000 t ), (2, 1100 t )	10	30	(15, 50 t ), (15, 80 t )
opm2	17	8	(4, 900 t ), (2, 1000 t ), (2, 1100 t )	10	30	(15, 50 t ), (15, 80 t )
opm5	17	8	(4, 900 t ), (2, 1000 t ), (2, 1100 t )	5	30	(15, 50 t ), (15, 80 t )
opm6	17	8	(4, 900 t ), (2, 1000 t ), (2, 1100 t )	5	30	(15, 50 t ), (15, 80 t )

Table 1: Characteristics of the instances [ 4 ] .

Figure 1: Feasibility ratio across confidence levels

\alpha

for all algorithms under uncertainty settings S1, S2, and S3 (top to bottom), for different values of

\kappa

Figure 2: Average chance-constraint margin across confidence levels

\alpha

under uncertainty settings S1, S2, and S3 (top to bottom) for different values of

\kappa

Instance	$\kappa$	$\alpha$	GSEMO (1)				NSGA-II (2)				NSGA-III (3)				MOEA/D (4)
			Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat
opm1	0.10	0.60	5876.9	81.5	26/30	2 (-) 3 (-) 4 (*)	5976.3	10.0	30/30	1 (+) 3 (*) 4 (+)	5973.7	14.5	30/30	1 (+) 2 (*) 4 (+)	5941.0	30.2	30/30	1 (*) 2 (-) 3 (-)
		0.70	5876.9	81.5	26/30	2 (-) 3 (-) 4 (*)	5976.3	10.0	30/30	1 (+) 3 (*) 4 (+)	5973.7	14.5	30/30	1 (+) 2 (*) 4 (+)	5941.0	30.2	30/30	1 (*) 2 (-) 3 (-)
		0.80	5883.3	81.6	24/30	2 (-) 3 (-) 4 (*)	5976.3	10.0	30/30	1 (+) 3 (*) 4 (+)	5973.7	14.5	30/30	1 (+) 2 (*) 4 (+)	5941.0	30.2	30/30	1 (*) 2 (-) 3 (-)
		0.90	5884.2	83.4	19/30	2 (-) 3 (-) 4 (*)	5977.0	7.9	30/30	1 (+) 3 (*) 4 (+)	5973.0	14.7	30/30	1 (+) 2 (*) 4 (+)	5941.0	30.2	30/30	1 (*) 2 (-) 3 (-)
	0.12	0.60	5876.9	81.5	26/30	2 (-) 3 (-) 4 (*)	5976.3	10.0	30/30	1 (+) 3 (*) 4 (+)	5973.7	14.5	30/30	1 (+) 2 (*) 4 (+)	5941.0	30.2	30/30	1 (*) 2 (-) 3 (-)

Table 2: Mean, standard deviation, statistical comparisons, and feasible-run count for the best feasible ore production under S1 across all

\kappa

values.

Instance	$\kappa$	$\alpha$	GSEMO (1)				NSGA-II (2)				NSGA-III (3)				MOEA/D (4)
			Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat
opm1	0.10	0.60	5877.0	78.9	27/30	2 (-) 3 (-) 4 (*)	5979.3	11.1	30/30	1 (+) 3 (*) 4 (+)	5979.3	12.3	30/30	1 (+) 2 (*) 4 (+)	5947.7	34.8	30/30	1 (*) 2 (-) 3 (-)
		0.70	5877.0	78.9	27/30	2 (-) 3 (-) 4 (*)	5979.3	11.1	30/30	1 (+) 3 (*) 4 (+)	5979.3	12.3	30/30	1 (+) 2 (*) 4 (+)	5947.7	34.8	30/30	1 (*) 2 (-) 3 (-)
		0.80	5877.0	78.9	27/30	2 (-) 3 (-) 4 (*)	5979.3	11.1	30/30	1 (+) 3 (*) 4 (+)	5979.3	12.3	30/30	1 (+) 2 (*) 4 (+)	5947.7	34.8	30/30	1 (*) 2 (-) 3 (-)
		0.90	5883.2	78.8	25/30	2 (-) 3 (-) 4 (*)	5980.0	9.1	30/30	1 (+) 3 (*) 4 (+)	5979.3	11.1	30/30	1 (+) 2 (*) 4 (+)	5947.7	34.8	30/30	1 (*) 2 (-) 3 (-)
	0.12	0.60	5877.0	78.9	27/30	2 (-) 3 (-) 4 (*)	5979.3	11.1	30/30	1 (+) 3 (*) 4 (+)	5979.3	12.3	30/30	1 (+) 2 (*) 4 (+)	5947.7	34.8	30/30	1 (*) 2 (-) 3 (-)

Table 3: Mean, standard deviation, statistical comparisons, and feasible-run count for the best feasible ore production under S2 across all

\kappa

values.

Instance	$\kappa$	$\alpha$	GSEMO (1)				NSGA-II (2)				NSGA-III (3)				MOEA/D (4)
			Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat	Mean	Std	Feas.	Stat
opm1	0.10	0.60	5879.2	77.9	26/30	2 (-) 3 (-) 4 (*)	5980.0	0.0	30/30	1 (+) 3 (*) 4 (+)	5975.0	13.1	30/30	1 (+) 2 (*) 4 (+)	5939.0	34.7	30/30	1 (*) 2 (-) 3 (-)
		0.70	5879.2	77.9	26/30	2 (-) 3 (-) 4 (*)	5980.0	0.0	30/30	1 (+) 3 (*) 4 (+)	5975.0	13.1	30/30	1 (+) 2 (*) 4 (+)	5939.0	34.7	30/30	1 (*) 2 (-) 3 (-)
		0.80	5879.2	77.9	26/30	2 (-) 3 (-) 4 (*)	5980.0	0.0	30/30	1 (+) 3 (*) 4 (+)	5975.0	13.1	30/30	1 (+) 2 (*) 4 (+)	5939.0	34.7	30/30	1 (*) 2 (-) 3 (-)
		0.90	5882.0	77.0	20/30	2 (-) 3 (-) 4 (*)	5977.3	8.7	30/30	1 (+) 3 (*) 4 (+)	5975.7	12.8	30/30	1 (+) 2 (*) 4 (+)	5939.0	34.7	30/30	1 (*) 2 (-) 3 (-)
	0.12	0.60	5879.2	77.9	26/30	2 (-) 3 (-) 4 (*)	5980.0	0.0	30/30	1 (+) 3 (*) 4 (+)	5975.0	13.1	30/30	1 (+) 2 (*) 4 (+)	5939.0	34.7	30/30	1 (*) 2 (-) 3 (-)

Table 4: Mean, standard deviation, statistical comparisons, and feasible-run count for the best feasible ore production under S3 across all

\kappa

values.

Instance	$\kappa$	$\alpha$	GSEMO				NSGA-II				NSGA-III				MOEA/D
			HV		IGD		HV		IGD		HV		IGD		HV		IGD
			Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
opm1	0.10	0.60	0.3585	0.3113	0.4509	0.2681	0.5382	0.0474	0.2784	0.0504	0.5277	0.0500	0.2853	0.0481	0.4112	0.0683	0.3688	0.0660
		0.70	0.3585	0.3113	0.4509	0.2681	0.5382	0.0474	0.2784	0.0504	0.5277	0.0500	0.2853	0.0481	0.4112	0.0683	0.3688	0.0660
		0.80	0.3839	0.3109	0.4285	0.2671	0.5382	0.0474	0.2784	0.0504	0.5277	0.0500	0.2853	0.0481	0.4112	0.0683	0.3688	0.0660
		0.90	0.3903	0.3181	0.4225	0.2748	0.5392	0.0436	0.2724	0.0411	0.5265	0.0553	0.2891	0.0510	0.4118	0.0675	0.3688	0.0660

Table 5: Mean and standard deviation of HV and IGD under S1 across all

\kappa

values.

Instance	GSEMO		NSGA-II		NSGA-III		MOEA/D
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
opm1	274.4	28.3	682.2	146.7	744.2	183.1	626.7	66.3
opm2	266.9	22.5	670.2	120.3	754.1	185.6	644.0	107.9
opm5	266.2	54.5	606.7	58.2	675.0	113.1	643.5	141.9
opm6	255.5	27.8	630.9	95.6	698.9	119.0	615.6	100.8

Table 6: Mean and standard deviation of computing time in seconds across all experimental configurations and independent runs.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Instance	$\kappa$	$\alpha$	GSEMO				NSGA-II				NSGA-III				MOEA/D
			HV		IGD		HV		IGD		HV		IGD		HV		IGD
			Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
opm1	0.10	0.60	0.3828	0.3269	0.5033	0.2993	0.5552	0.0523	0.2761	0.0333	0.5525	0.0503	0.2771	0.0321	0.4413	0.0747	0.3576	0.0635
		0.70	0.3828	0.3269	0.5033	0.2993	0.5552	0.0523	0.2761	0.0333	0.5525	0.0503	0.2771	0.0321	0.4413	0.0747	0.3576	0.0635
		0.80	0.3828	0.3269	0.5033	0.2993	0.5552	0.0523	0.2761	0.0333	0.5525	0.0503	0.2771	0.0321	0.4413	0.0747	0.3576	0.0635
		0.90	0.4176	0.3413	0.5329	0.3094	0.5520	0.0431	0.3384	0.0297	0.5569	0.0397	0.3348	0.0306	0.4409	0.0744	0.4188	0.0653

Appendix

Table 7: Hypervolume and IGD results under uncertainty setting S2 for all values of

\kappa

. Mean and standard deviation are computed over feasible recovered non-dominated fronts. HV and IGD use a configuration-specific empirical reference front and normalization. Bold values indicate the best mean performance for the corresponding metric and row.

Instance	$\kappa$	$\alpha$	GSEMO				NSGA-II				NSGA-III				MOEA/D
			HV		IGD		HV		IGD		HV		IGD		HV		IGD
			Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
opm1	0.10	0.60	0.3739	0.3074	0.4394	0.2757	0.5260	0.0515	0.2836	0.0419	0.5201	0.0587	0.2849	0.0462	0.4134	0.0643	0.3621	0.0503
		0.70	0.3739	0.3074	0.4394	0.2757	0.5260	0.0515	0.2836	0.0419	0.5201	0.0587	0.2849	0.0462	0.4134	0.0643	0.3621	0.0503
		0.80	0.3739	0.3074	0.4394	0.2757	0.5260	0.0515	0.2836	0.0419	0.5201	0.0587	0.2849	0.0462	0.4134	0.0643	0.3621	0.0503
		0.90	0.3845	0.2946	0.4185	0.2682	0.5382	0.0523	0.2521	0.0456	0.5205	0.0511	0.2687	0.0480	0.4134	0.0643	0.3520	0.0552

Appendix

Table 8: Hypervolume and IGD results under uncertainty setting S3 for all values of

\kappa

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On the use of evolutionary optimization for the dynamic chance constrained open-pit mine scheduling problem

Ishara Hewa Pathiranage, Aneta Neumann

Machine Learning and Optimisation, School of Computer Science and Information Technology, Adelaide University, Adelaide, Australia

Stochastic Optimization Chance-Constrained Optimization Multi-Objective Evolutionary Optimization

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Open-pit mine scheduling is a complex real-world optimization problem that involves uncertain economic values and dynamically changing resource capacities. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments. However, uncertainty and dynamic changes are often studied in isolation in real-world problems. In this paper, we study a dynamic chance-constrained open-pit mine scheduling problem in which block economic values are stochastic and mining and processing capacities vary over time. We adopt a bi-objective evolutionary formulation that simultaneously maximizes expected discounted profit and minimizes its standard deviation. To address dynamic changes, we propose a diversity-based change response mechanism that repairs a subset of infeasible solutions and introduces additional feasible solutions whenever a change is detected. We evaluate the effectiveness of this mechanism across four multi-objective evolutionary algorithms and compare it with a baseline re-evaluation-based change-response strategy. Experimental results on six mining instances demonstrate that the proposed approach consistently outperforms the baseline methods across different uncertainty levels and change frequencies.

Abstract

Stochastic Optimization

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