K-model based Mixture Design using D-optimal and A-optimal with Qualitative Factors

Objectives: To obtain a reliable approximation for the K-model in mixture experiments and design. Methods/Statistical Analysis: Here, the problem of mixture experiments, according to qualitative factors and finding A-optimal and D-optimal design for K-model is taking into account. Also, an improvement of Lee method is used to aim of this goal. In addition, a new procedure of Lee method for approximation of K-model is proposed. Moreover, illustrated examples are simulated in R software. Findings: It is demonstrated that the qualitative factor has a directly relation with A-optimal and D-optimal design. Such that, firstly, if the qualitative factor, on the region of factors, be a uniform design, then for A-optimal design, the trace of the inverse of the information matrix should be minimized. Secondly, for D-optimal design, maximization of the determination of information matrix is necessary. Moreover, in a product function, the dispersion function can be detached into 3 sections corresponding to the 2 marginal design. Application/Improvements: This research is using of an amount of convenient mixture design in engineering and manufacturing can be detached into 3 sections corresponding to the 2 marginal design. *Author for correspondence


Introduction
Recently, Mixture experiments have found a special importance in science and application. For instance, in food science, green manure, Agriculture and so on, one can see the role of mixture experiments 1,7 . For a better understanding, almost all of the cakes, are combined by alot of materials such as, flour, water, eggs, oil and etc. The amount of this material is very important to set the best product sometime due to increase or decrease of the materials. In most cases, The result is not desired the cakes taste, flavor and amount of puffing up depend on ingredients.
Indian Journal of Science and Technology, Vol 12 (8) The reason of delicacy of this product can be divided to two main sectors. The first sector is using the best material and the second one is this question that how long the material should be mix together 5 have suggested a general model for the linear combination of variables. The model is presented as follows: T j T f f j y E (1) Where, it shows the j-th level of a r-level qualitative factor and and ( ) Where the s C ' is additional constrains condition as in 6 defined. In this study, based on the result of 5 some concepts of A-optimal and D-optimal design is extended. The rest of the paper is as follows. In section 2 some preliminaries and some essential concepts to drive the tree of the information matrix of model (1) is presented. The main and analytical results are mentioned in section 3 such that according to the different condition of model (1), the A-optimal design for the mixture K-Model is finded. Finally conclusion and discussions are provided in section 4.

Preliminaries
Here, the general linear model is introduced by: And the design is measured by its information matrix worthy, which is demonstrated by:

A-optimal
A design is stated to be A -optimal if it minimizes the trance of the inverse of the information matrix. Works 4 Jack</author></authors></contributors><titles> <title>General equivalence theory for optimum designs (approximate theory and 3 gave us an effective way to check the A − optimality of arbitrary design ζ , and for a design ζ which is A − optimal if and only if: Let the general model (1) be written as: is the unit vector whose j − th component is equal to 1 and all others are 0 and ⊗ is used to Calculating the inverse matrix of ( ) According to computation of the inverse matrices of ( ) Therefore, the proof is completed.
Moreover, it also follows that, for ζ to be A − optimal, all the elements of η must be equal, i.e.

K-model Approximation based on A-optimal Designs
Theorem 1. Let the same assumptions of Lemma 1 is true and suppose that ( ) , , , ; It is better that for simplification, Equation (6) can be rewritten as , thus the following relations can be released.    We suppose that the q components mixture K-model symbol is demonstrated as: To fix ideas, we concentrate on the model which is given on We mostly consider three kinds of model which form as (1) given.
Likewise, we can change the two part of regression function as quantitative and qualitative factor, the model set as: However, there is no difference between fitting model (9) to model (11). In this work, qualitative and quantitative factors considered simultaneously, the design problems for estimation of the unknown parameters will be considered where it is assumed to have one qualitative factor with s levels.
In the model (10), the organization of information matrix is same as (13), then we obtain: where and We can explain the function (7) due to Theorem 1, Lemma 2 here, the condition is: Now, we should solve following equation to find the A −

D-optimal Designs for the K-model
Atkinson and Donev (1989) stated the BLKL-exchange algorithm due to the D-criterion for searching exact optimal designs with special block sizes. We can get lemma 3 easily because we have the information matrix in preliminaries.

Lemma 3.:
Suppose that τ be a product design with the marginal designs η and ξ on s X and X , respectively.
Then, the following equation of determinants is expressed for models (3) and (10): [ ] ) ) ( det( ) ) ( det( ) ( ) ) ( det( In the following, a connection of dispersion functions between the model (11) and (10) is derived for product designs. The determinant and inverse of a partitioned matrix can be obtained according to the formulas in Khuri (2003, pp. 35-6).