### G-2019-95

# Integrated lot sizing and blending problems

## Diego Fiorotto, Raf Jans, and Silvio Alexandre de Araujo

The standard blending problem consists of combining components to produce a final product with a given demand, while satisfying specific criteria with respect to the global blend and minimizing the total cost. The Bill-Of-Material (BOM) (or recipe) indicates which components are used and in which proportion. Typically, there is some flexibility in the planning process with respect to the proportion used for each of the components, where it may vary between a minimum and a maximum level instead of being fixed. This problem has been widely studied in a single period setting. However, the problem becomes more complex when we take into account a longer time frame. In such a case, demand for the final product occurs in several time periods, and both the final product and the components can be held in stock. In the integrated lot sizing and blending problem, the decisions relate to the production of the final product via the blending process, and the production (or procurement) of the components over an extended time horizon.

We propose mathematical formulations for this integrated problem. In a computational experiment, we analyse the impact of important parameters such as the level of flexibility in the BOM, the variance in the procurement cost among the components, and the variance of the proportion of the components in the total mix. Furthermore, we analyse the value of integration by comparing the solutions of the integrated models to the solutions of approaches that do not fully capture this integration such as a lot-for-lot approach, just-in-time models without inventory for the final product or components, and a hierarchical approach.

Published **December 2019**
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23 pages