Two Degree Becker Model for Mixture Design: Using D-optimal and A-optimal with Qualitative Factor

Objectives: To achieve an optimal approximation for two-degree Becker model in mixture design. Methods/Statistical Analysis: The problem of mixture design case, based on qualitative factors and finding A-optimal and D-optimal design for two-degree Becker model is investigated. With the aim of this issue, a generalization of Lee method is utilized. We proposed a new procedure of Lee method for approximation of Becker model. Moreover, simulation results are done in R software. Findings: There is a direct relation between qualitative factor and A-optimal and D-optimal design, such that, on the region of factors, if the qualitative factors have a uniform design then the trace of the inverse of information matrix is minimize for A-optimal design; and maximization of the determination of information matrix is essential for D-optimal design. Besides, for a product function, based on 3 sections corresponding to the 2-marginal design, the dispersion function can be detected. In addition, illustrated examples confirm the analytical results. Application/Improvements: The application of this work is to be used in engineering and manufacturing which need to an amount of convenient mixture design.


Introduction
Mixture experiment is one of the main procedures of manufactoring of a product and it has a vast range of application in industrial and technology. For instance, in Civil engineering 1 Chemical sciences 2 , medicine 3 and so on, one can see the role of mixture design in advance 4,5 . There are many forms of dietary supplements, for example, tablets, capsules, liquids, powders, and gels. Dietary supplements are different from drugs, and they are nonpatent drugs. The Food and Drug Administration (FDA) defined a dietary supplement as an alternative food containing essential nutrients like vitamins, minerals, and proteins 6 . Subsequently, the Nutrition Labeling and Education Act of 1990 added herb or nutritional substances to the definition. In the pharmaceutical industry, tablets are the most acceptable form for consumers in comparison with other oral dosage forms 7 . Tablet oral dosage has many advantages such as its ease of handling, chemical and physical stability, and portability. Furthermore, this type of dosage form ensures accuracy and consistency of dosages 8 . There are many examinations that can be done in order to maintain the physical qualities of the tablets, for example, hardness test, percentage of friability test, disintegration test, and dissolution test 9 . Tablets are mixtures of active ingredients and other excipients. Mixtures mean the sum of all the ingredients is 100%. There are many types of excipient with their own function in dosage formulation: diluents or fillers, binders, lubricants, glidants, antiadherents, disintegrates, colorants, and flavor or sweeteners. The mixture design statistical method is the most suitable method used in optimizing the tablet production process. The mixture design method is usually used in mixture formulation.
Here, there is the mean response at the j-th level of a s-level for qualitative factor as follows The fundamental objective of this study is to develop the results of the work 11 to the A-optimal designs of mixture model and 12,13 the D-optimal deign of mixture model. The rest of the artice is arranged as follows: In section 2, some basic preliminaries and some notations are provided. Also, calculation of the trace of information matrix of model (1) is provided therein. In addition, the main results are given in section 3. And we find the A-optimal design and D-optimal design for the wo degree Becker model based on different situations of model (1). Finally, section 4 provides concluding remarks.

Preliminaries
The general linear model given by

A-optimal
A design is defined to be A-optimal if it minimizes the trance of the inverse of the information matrix. The works 14,15 gave us an effective way to check the A-optimality of arbitrary designs ζ , and for a design ς which is A-optimal if and only if Let the general mode (1) be rewritten as where, s j R e ∈ is the unit vector whose j-th component is equal to 1 and all others are 0 and ⊗ is used  11 12 An arbitrary design on Ω can be expressed as where, η and j ξ are the marginal and the conditional designs on s χ and χ , respectively. Vol 12 (8) | February 2019 | www.indjst.org

Zahra Rasooli Berardehi and Chongqi Zhang
If ζ is supposed as a design production and it is presented by According to the result of 11 the information matrix of ζ will present by . Now, the following lemma can be obtained.

Lemma 1 16,17
For, an arbitrary design where η and ξ there are the conditional designs and the marginal on s x and x , respectively. Then one can has the following equation of trace for model (1).
By calculating the inverse matrices of ( ) And this completes the proof.
In particular, while the design ( )

D-optimal
In a design, if the determine of information matrix be maximizes the with that design, then it is as a D-optimal design. A useful way for checking the D-optimality is attention to this point that a design can be D-optimal if and only if ( ) ( ) ( ) 11 12 which, it is shown in (3) and this is same as information matrix with A-optimal design. In next section we will consider finding of the A-optimal and A-optimal designs for two degree Becker model under the this condition which:

Methodology
In this part, A-optimal and D-optimal method for the two degree Becker model are investigated.

A-optimal for the Two Degree Becker Model
For proving the A-optimality via the equivalence theorem, we can define the function as ; , , , .

Corollary 1 As a result of the lemma 1 and theorem 1, one has
And then by considering of the q components twodegree Becker model symbol as .., . , , ( ) ( Equivalently, the two part of regression function can be exchanged as quantitative and qualitative factors, so the model change as However, there isn't any main difference between model (6) to model (8). In this study, qualitative and quantitative factors are considered altogether, the problem of design to estimate the unknown parameters will be supposed where it is considered to exist one qualitative factor with s levels. The two degree Becker model mentioned that for models (6), (7) and (8) 6 × matrix, and its 1st, 2nd, , 6th rows should be T called the set of barycenter on the q-2 ension boundary. So, the design ξ can be define according to the models (6), (7) and (8)  2 ( ) 4 2 , and q I is the q q × identity matrix. According to these notations, the following Lemma can be expressed.

Lemma 2
For any design ξ such as (9) defined, the (7) and (8) Also, in the model (7) (7), then one can has Here, for finding of the A − optimal design * ζ for the model (7) and (8), solving the following equation is needed.

Conclusion
This study investigates the problem of mixture design case because of efficacy of mixture design in procedure of industrial experiences. In this regard, based on qualitative factors and finding A-optimal and D-optimal design for two-degree Becker model, the condition of production of mixture design is taken into account. It is worth to mention that, there is a direct relation between qualitative factor and A-optimal and D-optimal design. Such that, firstly on the region of factors, if the qualitative factors have a uniform design then the trace of the inverse of information matrix is minimize for A-optimal design. Secondly, maximization of the determination of information matrix is essential for D-optimal design. In addition, for a product function, based on three sections corresponding to the two marginal design, the dispersion function can be detected.