Uniform mixture design via evolutionary multi‐objective optimization

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Abstract

Design of experiments is a branch of statistics that has been employed in different areas of knowledge. A particular case of experimental designs is uniform mixture design. A uniform mixture design method aims to spread points (mixtures) uniformly distributed in the experimental region. Each mixture should meet the constraint that the sum of its components must be equal to one. In this paper, we propose a new method to approximate uniform mixture designs via evolutionary multi-objective optimization. For this task, we formulate three M-objective optimization problems whose Pareto optimal fronts correspond to a mixture design of M components (or dimensions). In order to obtain a uniform mixture design, we consider six well-known algorithms used in the area of evolutionary multi-objective optimization to solve M-objective optimization problems. Thus, a set of solutions approximates the entire Pareto front of each M-objective problem, while it implicitly approximates a uniform mixture design. We evaluate our proposed methodology by generating mixture designs in two, three, and up to eight dimensions, and we compare the results obtained concerning those produced by different methods available in the specialized literature. Our results indicate that the proposed strategy is a promising alternative to approximate uniform mixture designs. Unlike most of the existing approaches, it obtains mixture designs for an arbitrary number of points. Moreover, the generated design points are properly distributed in the experimental region.

Introduction

Design of experiments is a well-established methodology widely applied to experimental processes in the industry, process design, and science in general [1], [2], [3], [4], [5], [6], [7]. This approach has been found to be a powerful method to identify active and unimportant effects in an experimental process. However, many challenges arise from a practical standpoint, which have encouraged the development of different designs to meet practical needs. Mixture designs are a particular case of experimental designs subject to certain constraints. In a mixture, the independent components are proportions of different ingredients of a blend, and therefore, the sum of its components must be one. When the mixture design is only subject to the constraint that the components’ sum must be one, it is called standard mixture design. Examples of these methods are the Simplex-Lattice design [8] and the Simplex-Centroid design [9]. When the mixture design is subject to additional constraints, such as a maximum and/or minimum value for each component, it is referred to as Extreme-Vertices design (or constrained mixture design) [10].

The main goal of the uniform mixture design methods is to scatter the design points in the experimental region as uniformly as possible. Some authors have focused their studies on mixture designs from the viewpoint of classical optimal design [3], [8], [9], [11], [12]. This type of approaches aims to find an optimal distribution of points through an (M1)-dimensional simplex. However, as pointed out by some authors [6], [13], [14], [15], optimal design has several disadvantages:

  • 1

    An optimal design tends to distribute most of the design points on or near the experimental area’s boundary, leaving the interior mostly devoid of design points;

  • 2

    An optimal design is not a robust design to the assumed model; in most cases, the experimenter does not know the form of the model beforehand;

  • 3

    The high dimensionality of the mixtures makes it difficult for the existing methodologies to obtain an optimal design.

In order to address these drawbacks, researchers have developed diverse mixture designs from different nature [6], [16], [17], [18]. In this paper, we propose a new methodology for uniform mixture designs based on multi-objective evolutionary optimization. To this end, we must answer two questions: i) what multi-objective optimization problem should we solve? and ii) what evolutionary multi-objective algorithm should we use? In this research work, we formulate three M-objective problems whose Pareto optimal fronts correspond to a mixture design of M components, i.e., the Pareto front shapes describe a regular (M1)-dimensional simplex. Each M-objective optimization problem is solved using a multi-objective evolutionary algorithm. This way, a set of solutions approximates the entire Pareto front of each M-objective problem while a uniform mixture design is implicitly reached. In our experimental study, we identify the M-objective problem formulation for which the evolutionary algorithm searches for the best approximation to the entire Pareto front, i.e., the best approximation of a uniform mixture design. To validate our proposed approach, we generate uniform mixture designs in two, three, and up to eight dimensions, and we compare our results with respect to those produced by different uniform design methods available in the specialized literature. As we will see later on, our proposed approach is a promising alternative to approximate uniform mixture designs because it can create mixture designs for an arbitrary number of points distributed adequately in the (M1)-dimensional simplex.

The rest of the paper is organized as follows. Section 2 presents a review of different methods for mixture design. In Section 3, we introduce some basic concepts that will help to understand the rest of the paper. Section 4 introduces our proposed methodology. An experimental study of our proposed approach is presented in Section 5. Finally, our conclusion and some possible paths for future research are drawn in Section 6.

Section snippets

Related work

Uniform mixture design methods play an essential role in diverse areas of knowledge [1], [2], [3], [6]. According to their conceptual basis, we classify uniform mixture design approaches as follows.

Methods employing geometric concepts Scheffé [8] proposed the simplex-lattice design (SLD) technique in 1958.1

Mixture design

Experiments with mixtures have been very useful in different engineering and scientific areas. In experiments with mixtures, a response is assumed to depend on the proportions of the mixture components, not on the total amount of the mixture. Commonly, there are some additional constraints imposed on the components.

Formally, the constraints of the proportions (xi,i=1,2,M) in a mixture design with M components (or M dimensions) are stated as:i=1Mxi=1,xi0,fori=1,2,,M.Therefore, the

Our proposed approach

In this work, we formulated three M-objective optimization problems (MOPs). These MOPs have linear Pareto fronts, and their shapes describe a regular (M1)-dimensional simplex. Each solution in the Pareto front is a mixture, i.e., all components are nonnegative, and their sum is equal to one. Therefore, to produce a uniform mixture design, we need to solve any of the formulated MOPs.

Our methodology to approximate uniform mixture designs is called “Mixtures via Evolutionary Multi-objective

Performance assessment

The purpose of our proposed approach is to obtain a mixture design with design points as uniform as possible. In the above section, we saw that the Pareto front of the formulated MOPs is a set of nondominated points that describe a mixture design, i.e., each point in the Pareto front is a mixture. Therefore, uniformity in the Pareto front implies uniformity in the mixture design. Thus, we can use a performance indicator from the evolutionary optimization community to assess the distribution of

Conclusions

In this paper, we studied different methods to generate uniform mixture designs from the most classical approaches, such as the simplex-lattice design (SLD) and the simplex-centroid design (SCD), to the most recent approaches, such as the two-layered SLD, k-layer reference direction, and an approach based on low-discrepancy sequences. Perhaps, the most frequently used method for uniform mixture designs is the SLD. However, when the dimensionality of the mixture increases, the number of design

Declaration of Competing Interest

None.

Acknowledgment

The first author acknowledges support from PAPIIT project IA105918. The last author gratefully acknowledges support from CONACyT grant no. 2016-01-1920 (Investigación en Fronteras de la Ciencia 2016) and from a SEP-Cinvestav grant (proposal no. 4).

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