Create a fractional factorial design with mixture constraints
This work proposes a simple, generic and efficient approach to combine process and mixture factors in the same design by handing easily any mixture constraint in the frame of fractional factorial designs based on Rytz et al. (2017). Thus, computational functions were implemented in R allowing to quickly obtain fractional factorial designs with mixing constraints to define the experimental designs which were required to obtain products with different microstructures. Calculation functions have been compiled into an R package FrF2mix available on GitHub (https://github.com/adria-repo/FrF2mix).
To create the experimental designs combining process and mixing factors, easily dealing with any mixing constraint within the framework of fractional factorial designs, the maximum number of attempts authorized by the users were considered, with 2-levels for the identified process variables and the limit contents of the 3 ingredients considered in the mixing plan. Functions using the lexicon of the language R were written to develop a workflow, based on the methodology of Rytz et al. (2017), to (i) define experimental region according to an extreme vertices design in a constrained mixture space, (ii) build a pseudo-mixture design based on an orthogonal array and (iii) transform the pseudo-mixture design into final design using a single command.
A mixture experiment occurs when the response variable is a function of the relative proportions of ingredients in a mixture, rather than a function of the total amount of each component. Indeed, the proportions of components in a mixture must add to one. However, in many mixtures experiments each component can only be varied within specific lower and upper constraints. In this case, the experimental region is an irregular hyper-polyhedron rather than a simplex. Thus, in constrained experimental regions, experimental designs that consist of all the extreme vertices of the constrained region are recommended and can be optionally augmented by the centroids of the edges and facets (McLean and Anderson, 1966). Industrial food products can be described as processed mixtures of ingredients, but the complexity of product innovation and renovation lies in the fact that both process and mixture are highly multifactorial. Moreover, the qualities or characteristics of the product are influenced by process variables in addition to the proportions of the mixing components. Combining process and mixture factors in the same design is therefore a necessity. Scheffé (1963) proposed to make all combinations of a mixture design and a factorial design (i.e. Mixture × Process) but, in practice, the numbers of experiments are much too large. Box and Hau (2001) proposed a procedure that allows fractionating simultaneously the mixture and the process parts. Rytz et al. (2017) proposed to use all the advantages of the Box-Hau projection, while simplifying the adjustment (i.e. no need for a reference point) and making it therefore very generally applicable (i.e. homogeneous covering of asymmetrical experimental regions). Nevertheless, there are no computational functions for creating experimental designs combining process and mixing factors, easily dealing with any mixing constraint within the framework of fractional factorial designs.
R is a GNU-licensed free software programming language and software environment (R Core Team, 2022) primarily used for statistical computing as well as for dataviz. The R programming language is object oriented and provides objects, functions and operators to make using very natural. It is an extensible language, which allows the creation of new functions and extensions, such those being developed by the R community. Packages are sets of code and data that can be written by anyone in the R community. R packages can serve a number of uses. R users are familiar with the Comprehensive R Archive Network (CRAN), a massive repository that currently contains over 13,000 published R packages. But some packages can also be uploaded and downloaded from a version control and collaboration web platform (eg GitHub, GitLab). functions were written using the lexicon of the language R to develop a workflow, based on the methodology of Rytz et al. (2017), aiming to provide a user with a simple way to generate a fractional factorial design with mixture constraints for 3 ingredients. Via an association of the FrF2() and Xvert() functions of the FrF2 (Grömping, 2014) and mixexp (Lawson and Willden, 2016) packages, respectively, an experimental design is produced using a single command. Calculation functions have been compiled into an R package FrF2mix available on GitHub (https://github.com/adria-repo/FrF2mix).
In the context of the PLAN P project, there was a need to formulate different emulsions and foams according to a mixture design with process variables to obtain products of increasing complexity and different microstructures. The product domains in the project are fine-droplet liquid emulsions with high energy emulsification, foams resulting from a continuous expansion of emulsions, thick emulsions with low energy emulsification and spreadable emulsions in the form of fine pastes. To create the experimental designs of product domains, we considered the maximum number of attempts authorized by the partners, 2-levels for the identified process variables and the limit contents of the 3 ingredients considered in the mixing plan. functions were written using the lexicon of the language R to develop a workflow, based on the methodology of Rytz et al. (2017), to (i) define experimental region according to an extreme vertices design in a constrained mixture space, (ii) build a pseudo-mixture design based on an orthogonal array and (iii) transform the pseudo-mixture design into final design using a single command. These developed functions were used to set the experimental designs that were best suited to obtain products with different microstructures. For each protein ingredient selected to produce the different emulsions and foams, a maximum of 16 runs per design has been set to build the training database for a machine learning model.
