Incorporating grade uncertainty and stockpiling in stochastic open stope production scheduling optimization

The primary purpose of oil sands mine planning and waste management is to provide ore from the mine pit to the processing plant while containing the tailings in an efficient manner in-pit. Incorporating waste management in the mine plan is essential to maximize the economic potential of the mineral reserve and minimize waste management costs. However, spatial variability such as grade uncertainty results in ore tonnage variations, which leads to fluctuations in the quantity of ore to be processed and waste to be managed. This paper investigates the application of a stochastic mixed integer linear programming (SMILP) on oil sands mine planning to integrate bitumen grade uncertainty and waste management. Sequential Gaussian simulation is employed to quantitatively model the spatial variability of bitumen grade in the oil sands deposit. Multiple simulated orebody models are used as inputs for the SMILP model to generate optimal mine plans in the presence of grade uncertainty. The results demonstrate that the SMILP schedule generates 14% and 17% improvements in net present value compared to the E-type and ordinary kriging schedules, respectively. These results indicate that the SMILP model is a robust tool for optimizing stochastic integrated oil sands production schedules and waste management.

Conventional mine planning approaches use an estimated orebody model as input to generate optimal production schedules. The smoothing effect of some geostatistical estimation methods cause most of the mine plans and production forecasts to be unrealistic and incomplete. With the development of simulation methods, the risks from grade uncertainty in ore reserves can be measured and managed through a set of equally probable orebody realizations. In order to incorporate grade uncertainty into the strategic mine plan, a stochastic mixed integer programming (SMIP) formulation is presented to optimize an underground cut-and-fill mining production schedule. The objective function of the SMIP model is to maximize the net present value (NPV) of the mining project and minimize the risk of deviation from the production targets. To demonstrate the applicability of the SMIP model, a case study on a cut-and-fill underground gold mining operation is implemented.

Mineral deposits are the main assets for the mining industry. Mineral deposits are estimated based on the findings of exploration drilling. Complex host geology with variable grades and geological controls increases difficulty in resource estimation. In these situations, volume (tonnage) and grade are often over- or underestimated, resulting in inaccurate mine plan that leads to costly financial decisions. In this study, a multiple-point geostatistical method, namely Single Normal Equation Simulation (SNESIM) was applied to generate equiprobable orebody models for a copper deposit from Africa that helps to analyze the uncertainty of ore tonnage of the deposit. The grade uncertainty was evaluated by generating multiple realization of grade models using sequential Gaussian simulation within each equiprobable orebody models. The results are validated by generating the marginal distribution, and two- and three-point statistics. In addition, a comparative study is performed for the deterministic version, the stochastic version with grade uncertainty, and the stochastic version with volume and grade uncertainty. The results show that the orebody model with the maximum volume is 4.8% more than the average volume and the minimum volume is 5.1% less than the average volume. The grade simulation results demonstrate that the average grade for all simulations is 3.89%, but average grade for different simulations varied from 3.6% to 4.1%. The results also show that the volume and grade uncertainty model overestimated the orebody volume compared to the conventional orebody volume. The long-term production schedule is generated taking into account the volume and grade uncertainties from the orebody models, and satisfying mine production capacity and xi processing capacity constraints. The production schedule results for the volume and grade uncertainty-based model are compared to the production schedule generated from deterministic orebody model, and grade uncertainty-based model. The results demonstrated that the incorporation of both the volume and grade uncertainty significantly reduces the risk of deviation from the target. The results also show that incorporation of volume and grade uncertainty increases the net present value (NPV) of mining project, when compared to the mine plan generated from the deterministic model and stochastic model with only grade uncertainty. The results show that the production schedule generates high revenue over wide range of initial assumptions and the expected NPV is 3% higher than the deterministic version. A sensitivity analysis was also performed to understand the effect of penalty factor for deviating the constraints.

The inclusion of grade uncertainty for multivariate mineral deposits is of great importance for the correct management of subsequent decisions involved in mining planning. Mapping grade uncertainties allows maximization of profit and resource extraction. In this article, the co-simulation turning band algorithm is applied with the aim of predicting multivariate grade uncertainties. Moreover, a probabilistic analysis in long term mining sequencing is proposed in order to select the best given grade scheduling uncertainty derived from the simulations. A case study in a phosphate mine shows that the correlation of co-simulated variables honors the original data and there is an improvement in the project by an increase in Net Present Value (NPV) planning considering grade uncertainties. A comparison is performed with the results derived from the selected schedule and the results using the model based on kriged grades.

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)

Due to uncertain nature of grade in ore deposits, considering uncertainty is inevitable in geological modelling of resources and mine planning. In other words, uncertainty in grade of mineralized materials, is one of the most significant parameters need attention in mine planning. In this paper, a comparative procedure utilizing Sequential Gaussian Simulation (SGS) and traditional Ordinary Kriging (OK) was applied in an iron ore mine, and the influence of ore grade uncertainty in mine planning was investigated. It was observed that grade distribution, resulted from the SGS is almost identical to that of the real exploration data as compared to the OK method. Also it is emphasized that uncertainties including ore grade of deposit would significantly affect the technical and financial aspects of plans. Comparison shows that the simulation-based ultimate pits exhibits less risk in deviating from quantity and quality targets than traditional approach based on a single orebody model obtained by OK method. Using SGS method, there was an increase in the value of net present value of mine plans.

Deep Reinforcement Learning for Managing Platelets in a Hospital Blood Bank

We consider the problem of scheduling under demand uncertainty a multiproduct batch plant represented through a state−task network. Given a scheduling horizon consisting of several time periods in which product demands are placed, the objective is to select a schedule that maximizes the expected profit. We present a multistage stochastic mixed integer linear programming (MILP) model, wherein certain decisions are made irrespective of the realization of the uncertain parameters and some decisions are made upon realization of the uncertainty. To overcome the computational expense associated with the solution of the large-scale stochastic multistage MILP for large problems, we examine an approximation strategy based on the solution of a series of a two-stage models within a shrinking-horizon approach. Computational results indicate that the proposed approximation strategy provides an expected profit within a few percent of the multistage stochastic MILP result in a fraction of the computation time and provid...

This paper describes a cardinality constrained network flow structure whose special characteristics are used to analyse different risk aspects under an environment of uncertainty. The network structure developed is a suitable alternative to support financial planning and many other decision-making problems with limited resources. By setting a diversification level, we can manage systematic and non-systematic risks under a stochastic mixed integer linear programming framework. A dual decomposition method, Progressive Hedging (PH), is applied to more efficiently accommodate instances with large numbers of scenarios. We studied the impact of the level of the diversification on transaction costs and considered different factors that influence the performance of the algorithm. In particular, a Lagrangian bound is embedded to enhance the capacity of the method. Numerical results show the effectiveness of the proposed decision support approach.

Turbine investment optimisation for energy recovery plants by utilising historic steam flow profiles

The conventional approach to mine planning is to use a single estimated orebody model as the basis for production scheduling. This approach, however, does not consider grade uncertainties associated with grade estimation. These uncertainties have a significant impact on the net present value (NPV) and can only be accounted for when modelled as part of the production scheduling optimisation problem. In this research, a set of equally probable simulated orebodies generated through Sequential Gaussian Simulation is used as input to a stochastic optimisation model solved with genetic algorithm (GA). Grade variability is considered as part of the stochastic model. The problem definition and resource constraints are formulated and optimised using a specially designed mining-specific GA. This GA is employed to handle partial block processing through a specialised chromosome encoding technique resulting in near-optimal solutions. Two case studies are presented which compare results from the stochastic model solved with GA (SGA) and a Stochastic Mixed Integer Linear Programming (SMILP) model solved with CPLEX. For the second case study, while the SMILP model was at an optimality gap of 101% after 28 days, the SGA model generated an NPV of $10,045 M at 10.16% optimality gap after 1.5 h.

Optimal self-scheduling of home energy management system in the presence of photovoltaic power generation and batteries

برنامه‌ریزی تولید بلند‌مدت در معادن روباز یک امر بسیار حیاتی در برنامه‌ریزی معدن است و توزیع جریان نقدینگی را در سراسر عمر معدن مشخص می‌کند. هدف برنامه‌ریزی، بیشینه‌کردن ارزش خالص فعلی با در نظر گرفتن همه محدودیت‌های عملیاتی از قبیل شیب، آمیختن عیارهای مختلف، تولید ماده معدنی و ظرفیت استخراج است. عدم قطعیت‌های مرتبط با داده‌های مدل، نقش به سزایی در بهینه‌سازی برنامه‌های تولید بلند‌مدت دارند. در میان عدم قطعیت‌ها، عدم قطعیت عیار، سهم عمده‌ای را ایفا می‌کند. در این مقاله مدل‌های ترکیبی به وسیله روش آزادسازی لاگرانژی (LR)، روش آزادسازی لاگرانژی تعمیم‌یافته (ALR) و الگوریتم کرم شب‌تاب (FA) برای حل مساله برنامه‌ریزی تولید بلند مدت معادن روباز با فرض قطعیت و همچنین، با در نظر گرفتن عدم قطعیت عیار ارایه شده‌اند. الگوریتم کرم شب‌تاب برای به روزرسانی ضرایب لاگرانژ مورد استفاده قرار گرفته شده است. رویکردهای جدید پیشنهاد شده با نتایج روش‌های ترکیبی حاصل از آزادسازی لاگرانژی و آزادسازی لاگرانژی تعمیم‌یافته با الگوریتم ژنتیک (GA) و روش سنتی زیرگرادیان (SG) مقایسه شده‌اند. برای حل و اعتبارسنجی مدل به دست‌آمده، معدن سنگ آهن چادرملو به عنوان مورد مطالعاتی مناسب، در نظرگرفته شده است. نتایج حاصل از مطالعه موردی نشان می‌دهد که استراتژی‌ ترکیبی ALR-FA می‌تواند راه‌حل بهینه را نسبت به روش‌های دیگر ارایه کند؛ به‌طوری‌که، در طول یک دوره زمان‌بندی دوازده ساله، میانگین ارزش خالص با استفاده از روش ترکیبی پیشنهادی 11/20 درصد بیشتر از روش سنتی موجود است. همچنین، سرعت CPU از مدل پیشنهادی، 7/4 درصد بیشتر از دیگر روش‌ها حاصل شد.

This work investigates the design optimization of aggregated energy systems (multi-energy systems, microgrids, energy districts, etc.) with (N-1)-reliability requirements. The problem is formulated as a two-stage stochastic Mixed Integer Linear Program which optimizes design (first stage variables) and operation variables (second stage variables) simultaneously considering a set of typical and extreme days. The analysis proposes and compares different approaches to include the (N-1) reliability requirement in the optimization problem. Moreover, the paper proposes two effective decomposition algorithms to solve the large-scale Mixed Integer Linear Program suitable for design problems with and without (N-1) reliability requirements. Depending on the instance, such decomposition algorithms allow reducing the computational time by one or more orders of magnitude (from days to a few hours, in the worst cases tested in this work). The proposed methodology is tested to design the aggregated energy system for a real case study considering both a grid-connected and off-grid installation. Results indicate that the actual reliability of the design solutions depends by the profiles of energy demand and renewable production considered in the failure scenarios included in the design problem. Including N-1 reliability requirements causes an increase in the total annual cost in the range 15–20%, due to the increase in capital costs.

An inexact stochastic mixed integer linear semi-infinite programming (ISMISIP) model is developed for municipal solid waste (MSW) management under uncertainty. By incorporating stochastic programming (SP), integer programming and interval semi-infinite programming (ISIP) within a general waste management problem, the model can simultaneously handle programming problems with coefficients expressed as probability distribution functions, intervals and functional intervals. Compared with those inexact programming models without introducing functional interval coefficients, the ISMISIP model has the following advantages that: (1) since parameters are represented as functional intervals, the parameter’s dynamic feature (i.e., the constraint should be satisfied under all possible levels within its range) can be reflected, and (2) it is applicable to practical problems as the solution method does not generate more complicated intermediate models (He and Huang, Technical Report, 2004; He et al. J Air Waste Manage Assoc, 2007). Moreover, the ISMISIP model is proposed upon the previous inexact mixed integer linear semi-infinite programming (IMISIP) model by assuming capacities of the landfill, WTE and composting facilities to be stochastic. Thus it has the improved capabilities in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost and (2) addressing tradeoffs among environmental, economic and system reliability level.