# Order of Addition in Mixture‐Amount Experiments

## TL;DR

This paper introduces a new method for designing experiments where both the proportions and order of adding mixture components matter.

## Contribution

The paper proposes using the Threshold Accepting algorithm to optimize experimental designs for mixture-amount experiments with order-of-addition effects.

## Key findings

- The Threshold Accepting algorithm effectively selects optimal experimental runs for mixture-amount designs.
- G-efficiency criterion ensures precise and unbiased estimation of model parameters.
- FDS plots visually assess prediction capabilities across the design space.

## Abstract

In a mixture experiment, we study the behavior and properties of m mixture components, where the primary focus is on the proportions of the components that make up the mixture rather than the total amount. Mixture‐amount experiments are specialized types of mixture experiments where both the proportions of the components in the mixture and the total amount of the mixture are of interest. In this paper, we consider an Order‐of‐Addition (OofA) mixture‐amount experiment in which the response depends on both the mixture amounts of components and their order of addition. Full mixture OofA designs are constructed to maintain orthogonality between the mixture‐amount model terms and the effects of the order of addition. But the number of runs in such full OofA designs increases as m increases. We employ the Threshold Accepting (TA) Algorithm to select an n‐row subset from the full OofA mixture design that maximizes G‐optimality while minimizing the number of experimental runs. Further, the G‐efficiency criterion is used to assess how well the design supports the precise and unbiased estimation of the model parameters. These designs enable the estimation of mixture‐component model parameters and the order‐of‐addition effects. The Fraction of Design Space (FDS) plot is used to provide a visual assessment of the prediction capabilities of a design across the entire design space.