## Rancangan D-Optimal pada Mixture-Process Variable dengan Pendekatan Petak Terbagi Menggunakan Algoritma Genetika

##### Date

2022-01##### Author

Juniar, Dea Handayani

Wigena, Aji Hamim

Syafitri, Utami Dyah

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Experimental design is one of the fields of statistical science that is often used in the Industrial sector, especially in manufacturing product compositions. Experimental designs that are often used in the manufacture of product compositions are mixture experiments. A mixture experiment is an experiment that the factors are the components of a mixture. The response is assumed to depend only on the relative proportions of each component and not the total amount of the mixture (Cornell, 2002).. The unique feature in the mixture experiment is the mixture constraint. The proportion of each component must be between zero and one and sum to unity (Montgomery 2001).. In practice, Industrial experiments determining the proportion of product components are usually carried out by trial and error. However, trial & error is not efficient for experiments with many components. Therefore, designing an experiment can help researchers determine the right combination of product components and are more structured.
In some cases, the response depends not only on the relative proportions of each component but also on process variables. These process variables are not part of the mixture, but if the conditions or levels are changed, these changes will affect the response of interest. This combination of mixture experiment and process variables is called Mixture Process Variable (MPV). In practice, the composition of the mixture experiment will run for each level of the process variables. The level of process variables may be hard to change from one level to another. Therefore, it causes a limitation in randomization. A split-plot approach can be an option to solve the problem, where the whole-plot is the process variables, and the sub-plot is the mixture component. The whole plot is a factor with hard to change, while the subplot is a factor with easy to change from run to run. In addition, the impact of MPV is a large number of experimental runs because the relative proportions of each component must be executed at each level of the process variable. Regularly the number of runs in an experiment must be limited due to cost, time, and resource constraints. However, choosing an optimal design with limited runs will ensure efficiency in such cases.
The optimal design is part of the experimental design that estimates parameters without bias and minimum variance. Practically, the optimal design can reduce the cost and time in the experiment. The optimal design is searching for a good design based on a specific criterion. Finding an excellent optimal design concerning some optimality criteria can be difficult. In this study, a genetic algorithm (GA) was developed to find the optimal design. A genetic algorithm maintains a population of candidate solutions to a problem. Then choose the candidate who has the most suitable criteria to solve the problem. The selection criteria used is the D-optimality criterion based on parameter prediction. The D-optimal design is obtained by minimizing the variance value of the estimated model parameters. The case study was used to create the best recipe of two food recipes to accomplish this research with consisted of three mixture components and a process variable with three levels. Each material in each case has specific constraints resulting in an increasingly complex design area. The research has two main aims: (i) To build a genetic algorithm in R software and obtain a D-optimal design for a mixture-process variable (MPV) with a split-plot approach, and (ii). To compare the results of the genetic algorithm in R software with the coordinate-exchange algorithm in the JMP Software using D-efficiency.
In conclusion, the genetic algorithm thrived on finding the D-optimal MPV design with a split-plot approach. The strategies used in the genetic algorithm were blending, crossover, mutation, sign change, and zero genes. Blending and mutation operators used less probability of success (α) than other operators to obtain more stable D-optimal MPV designs. Compared with the coordinate exchange algorithm, the genetic algorithm can provide more efficient results assuming a linear process variable with three levels.