Order-of-addition in mixture and component-amount designs in orthogonal blocks

Mixture experiments often involve process variables, where orthogonal block designs allow independent estimation of mixture component parameters and process variable parameters. Traditionally, the focus in mixture experiments has been on the proportions of components rather than the order in which they are added. This paper introduces the concept of order-of-addition (OofA) effects in mixture blends within orthogonal blocks, where the response depends on both the proportions of the components and their sequence of addition. We also extend this framework to mixture-amount experiments in orthogonal blocks, incorporating OofA effects into component-amount designs. We first construct mixture and component-amount designs in non-orthogonal blocks, including pairwise ordering variables. To improve efficiency, we then employ the Threshold Accepting (TA) algorithm to reduce the number of runs and generate G-optimal orthogonal designs in two blocks for both mixture and component-amount experiments. In accordance, the proposed designs allow estimation of primary mixture or component-amount parameters, as well as their interactions with OofA effects, without confounding with block effects. The efficiency of the methodology is demonstrated through illustrative examples from pharmaceutical applications using simulated responses. The findings highlight the potential of orthogonally blocked OofA designs to provide a more comprehensive understanding of mixture and component-amount systems in biomedical and pharmaceutical research.