A novel approach to robust optimization of coal blending operations is proposed. While many rigorous mathematical approaches have been developed for this problem in the past, most neglect the uncertainty associated with both the inherent coal quality attributes as well as the sampling and measurement protocols used to measure those attributes. This uncertainty may lead to the imprecise estimation of quality values and force operators to adopt a conservative blend strategy based on safety factors and simple heuristics. One promising approach to rigorously address this shortcoming is through robust optimization (RO), a unique approach to handle data with uncertain and unspecified statistical distributions. In the current work, we first derive a deterministic optimization problem using the premium/penalty structure in typical coal sales contracts. This formulation is then translated to a robust formulation and solved as a Mixed Integer Nonlinear Program (MINLP). A numerical example is presented to demonstrate the utility of the method and to compare the deterministic solution, as the most optimistic and risky blending plan, with the RO solution, as the more pessimistic, yet robust, plan. The proposed strategy can help decision-makers achieve the economic and strategic goals of the operation at acceptable risk levels.

The design of cell-based flotation circuits is often completed in two distinct phases, namely circuit structure identification and equipment sizing selection. While recent literature studies have begun to address the implications of stochastic analysis, industrial practice in flotation circuit design still strongly favors the use of deterministic metallurgical modeling approaches. Due to the complexity of the available mathematical models, most flotation circuit design techniques are constructed based on deterministic models. Neglecting the impact of various sources of uncertainty may result in the identification of circuit solutions that are only optimal in a narrow region of specific operating scenarios. One promising strategy to address this shortcoming is through the Sample Average Approximation (SAA) methodology, a stochastic approach to handling uncertainty that has been widely applied in other disciplines such as supply chain and facility location management problems. In this study, a techno-economic optimization algorithm was formulated to select the optimal size and number of flotation cells for a fixed circuit structure while considering potential uncertainty in several input parameter including feed grade, kinetic coefficients, and metal price. Initially, a sensitivity analysis was conducted to screen the uncertain parameters. After simplifying the optimization problem, the SAA approach was implemented to determine the equipment configuration (i.e., cell size and number) that maximizes the plant’s net present value while considering the range of potential input values due to parameter uncertainty. The SAA methodology was found to be useful in analyzing uncertainty in flotation kinetics; however, the approach did not provide a useful means to assess the influence of uncertainties in ore grade and metal price, as these values are not significant in determining equipment size but rather influence the optimal circuit structure, which was not considered in this study. Results from an application example indicate that the SAA approach produces optimal solutions not initially identified in a deterministic optimization, and these SAA solutions tend to provide greater robustness to uncertainty and variation in the flotation kinetics.