(664b) Scheduling and Feed Quality Optimization for Raw Materials in the Metals Industry

Yingkai Song^*, Brenno C. Menezes^*, Pablo Garcia-Herreros^§, Ignacio E. Grossmann^*

^*Department of Chemical Engineering. Carnegie Mellon University. Pittsburgh, PA 15213, USA.

^§Research, Development & Innovation. Aurubis AG. Hamburg, Germany.

Abstract

We address the scheduling and feed quality optimization for processing of raw materials in non-ferrous metal industries. The model considers scheduled arrival of ships carrying solid concentrates, internal logistics, blending operations, and concentrate smelting. The raw materials are initially received at the port, where they are stored at warehouses and a first mixing step takes place. The mixed materials are sent to the processing facility through a river transportation system. Once the mixtures are received at the production facility, several conveyor belts deliver them to the smelter together with other non-concentrate materials. The proposed problem is similar to the classic crude oil scheduling optimization [1], although there are additional challenges to solve the logistics and quality problems that we describe below.

We develop a multi-period Mixed-Integer Non-Linear Programming (MINLP) model that includes several special features. First, special mixing rules are modeled for small ships and conveyor belts, which introduce non-convex bilinear terms. Second, variations in the concentration of the mixture are undesirable in the smelting process for operational stability. For this reason, penalties are considered for variations in the concentration; they provide a balance between profit and operational stability. In addition to upper bounds for concentrations in the final feed blend, which are common in crude oil scheduling problem, interdependency constraints based on individual element concentrations are considered in our model.

The large scale non-convex MINLP is prohibitively expensive to be solved by global optimization solvers. Furthermore, the non-convexities can make a local non-linear solver fail in finding feasible solutions or lead to suboptimal solution. Thus, we implement a decomposition scheme for the MINLP formulation representing the quantity-logic-quality phenomena [2,3]. We propose a two-stage solution approach that involves solving a Mixed-Integer Linear Problem (MILP) that relaxes the non-convex constraints, and a Non-Linear Problem (NLP) that fixes discrete decisions. Several solution strategies, such as rolling horizon, convex relaxations [4], and heuristics for finding feasible solutions have been implemented to avoid infeasibilities and to reduce the gap between the MILP and NLP problems. The examples show that the proposed solution strategy yields near optimal solutions with reasonable computational effort.

[1] Lee, H., Pinto, J., Grossmann, I., & Park, S. (1996). 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.

[2] B.C. Menezes, J.D. Kelly, I.E. Grossmann, 2015, Phenomenological Decomposition Heuristic for Process Design Synthesis in the Oil-Refining Industry. Comput Aided Chem Eng, 37, 1877-1882.

[3] J.D. Kelly, B.C. Menezes, F. Engineer, I.E. Grossmann, 2017, Crude-oil Blend Scheduling Optimization of an Industrial-Sized Refinery: a Discrete-Time Benchmark. Comput Aided Chem Eng, In Press.

[4] Andrade, T., Ribas, G., & Oliveira, F. (2015). A Strategy Based on Convex Relaxation for Solving the Oil Refinery Operations Planning Problem. Industrial & Engineering Chemistry Research, 55(1), 144-155