(26a) Optimal Scheduling of Copper Concentrate Operations Under Uncertainty

Authors: 
Cheng, P., Carnegie Mellon University
Garcia-Herreros, P., Aurubis A.G.
Lalpuria, M., Aurubis A.G.
Grossmann, I., Carnegie Mellon University
The scheduling of copper concentrate operations is a complex optimization problem. Copper concentrates need to be transferred from ports to a pre-blending unit, to bins and finally to the smelter, where they are transformed into copper matte. All assignments and timings of these transfer operations need to be determined respectively. Meanwhile, there are several by-products for the smelting process, i.e. iron slag and sulfur dioxide, and these by-products along with copper matte all have their own quality requirements, which sets more restrictions on the schedule. In addition, the elemental composition of concentrates fluctuates within certain bounds because of the complex upstream mining process. This uncertainty is crucial to quality requirements, and may cause the deterministic optimal schedule to be sub-optimal or even infeasible. Although the scheduling of copper concentrate operations shares some common features with crude oil scheduling problem, such as decision types and process topology, only the latter is well studied in the process systems engineering field. Discrete-time and continuous-time formulations for crude oil scheduling problems have been addressed by [1, 2]. For the scheduling of copper concentrate operations, Song et al. [3] developed a discrete-time scheduling model to address the deterministic problem without considering the impact of uncertainty. This problem has not been addressed before using continuous-time model and taking the uncertainty in elemental composition into consideration.

In this work, we address the scheduling of copper concentrate operations using continuous-time formulation, and deal with the uncertainty in elemental composition using both robust optimization and flexibility analysis. Robust optimization [4] follows the idea of optimizing against adverse realization of the uncertainty while guaranteeing the feasibility of the solution. Li et al. [5] reviewed methods for scheduling under uncertainty, including several robust optimization approaches. Flexibility analysis [6] is an uncertainty modeling technique that was originally used for plant design under uncertainty and has been developed in the process systems engineering community for more than thirty years. As shown in a recent paper by Zhang et al.[7], flexibility analysis can have either identical or even better results than robust optimization for linear systems, although it is more time consuming.

To address the scheduling of copper concentrate operations under uncertainty, we use a multi-operation sequencing (MOS) model to formulate the deterministic scheduling problem, and utilize non-overlapping operations to tighten the formulation. Based on the deterministic model, we develop a robust MOS model, within which the robust optimization is embedded into the MOS model to formulate the worst case of the uncertainty, and the flexibility test is used to obtain the values of uncertain parameters in the corresponding worst case. The robust MOS model can be further extended to a multi-objective robust MOS model to allow the violations of quality requirements when a large range of uncertainty is included in the model. The three models are all non-convex Mixed-Integer Nonlinear Programming (MINLP) problems, which can be solved by a tailored two-step MILP-NLP decomposition strategy. An industrial case with 14 concentrates over a 15-day time horizon shows that all three models achieve near global optimal solutions within reasonable time with the decomposition strategy.

[1] Heeman Lee, Jose M Pinto, Ignacio E Grossmann, and Sunwon Park. Mixed-integer linear programming model for refinery short-term scheduling of crude oil unloading with inventory management. Industrial & Engineering Chemistry Research, 35(5):1630–1641, 1996.

[2] Sylvain Mouret, Ignacio E. Grossmann, and Pierre Pestiaux. Time representations and mathematical models for process scheduling problems. Computers and Chemical Engineering, 35(6):1038–1063, 2011.

[3] Yingkai Song, Brenno C. Menezes, Pablo Garcia-Herreros, and Ignacio E. Grossmann. Scheduling and Feed Quality Optimization of Concentrate Raw Materials in the Copper Refining Industry. Industrial and Engineering Chemistry Research, 57(34):11686–11701, 2018.

[4] Dimitris Bertsimas, David B Brown, and Constantine Caramanis. Theory and applications of robust optimization. SIAM review, 53(3):464–501, 2011.

[5] Zukui Li and Marianthi Ierapetritou. Process scheduling under uncertainty: Review and challenges. Computers and Chemical Engineering, 32(4-5):715–727, 2008.

[6] Ignacio E. Grossmann, Bruno A. Calfa, and Pablo Garcia-Herreros. Evolution of concepts and models for quantifying resiliency and flexibility of chemical processes. Computers and Chemical Engineering, 70:22–34, 2014.

[7] Qi Zhang, Ignacio E Grossmann, and Ricardo M Lima. On the relation between flexibility analysis and robust optimization for linear systems. AIChE Journal, 62(9):3109–3123, 2016.