D-optimal Designs for Quartic Polynomial Model for Mixture Experiment with Three and Four Ingredients

Abstract

This article considers a classical optimal design problem of finding saturated D-optimal designs for the quartic polynomial model for mixture experiments. The necessary and sufficient conditions of the optimality criterion are confirmed by using the corresponding equivalence theorem.

AMS Subject Classification: 62K05

Keywords

Saturated design D-optimal design quartic model mixture experiment equivalence theorem

1. Introduction

Mixture experiments have been extensively used in different disciplines such as agriculture, pharmaceutical science, chemical science and nutritional science. For the $q$ components mixture system, the response is assumed to be a function of component proportions denoted by $x_{1}, x_{2}, \dots, x_{q}$ . Further, in a mixture experiment, it is also assumed that the response of interest does not depend upon the amount of the mixtures. For example, in the case of a gasoline blend, where the response of interest is the gasoline research octane number at 2.0 g, which depends only on the proportion of the ingredients present in the mixture, that is, butane, alkylate, light straight run, reformate, catalytically cracked and others (see Snee and Marquardt ^[1]). The set containing the mixture becomes a $(q - 1) -$ dimensional simplex and can be represented as:

ζ = \{x = {(x_{1}, x_{2}, \dots, x_{q})}^{'} \in R^{q} | \sum_{i = 1}^{q} x_{i} = 1, 0 \leq x_{i} \leq 1, i = 1, 2, \dots, q\}

(1.1)

In general, the construction of optimal designs aims at making the predicted response closer to the mean response over a particular region based on a specific criterion. In the literature, different optimality criteria have been discussed, such as D-optimality, A-optimality, E-optimality, R-optimality and V-optimality. The works of Kiefer and Wolfowitz^[2] and Kiefer^[3] are the pioneering contributions related to optimal designs for mixture experiments.

The research work on optimal designs for mixture experiments continues to attract the attention of many learned scholars to carry out their research work in this particular domain. Recently, many articles were published that discuss the construction of optimal designs for various models of mixture experiments, such as Pal and Mandal,^[4] Panda and Sahoo^[5–7] and Panda.^{[8, 9]} For more detailed information on optimal designs for mixture experiments, readers may refer to the work of Sinha et al.^[10]

The canonical polynomial models proposed by Scheffè^[11] are widely used for analyzing mixture data, especially Scheffè’s quadratic model is quite useful for analyzing agricultural and industrial data. However, there are also higher-degree canonical polynomial models, such as cubic and quartic polynomial models, for mixture experiments defined in the literature. One should prefer cubic and quartic polynomial models in comparison to quadratic models if the interest is to model the curvature in the interior of a factor space. For more details, one can refer to the work of Cornell.^[12]

Kiefer^[3] obtained D-optimal designs for Scheffè’s models of degrees one, two and three. He proved that a saturated design that allocates equal weight to each vertex of the simplex region is the D-optimal design for the first-degree Scheffè’s polynomial model. Further, he also showed that the simplex-lattice design that allocates equal weight to each support point is D-optimal for the second-degree Scheffè’s polynomial model. For the three-component mixture, he also obtained saturated D-optimal designs for the full cubic model, the cubic model without a three-way effect, and the special cubic model. Subsequently, Farrell et al.^[13] and Lim^[14] derived the D-optimal designs for the general cubic polynomial model with two and three mixture components, respectively. Mikaeili^{[15, 16]} obtained D-optimal designs for the cubic model without the three-way effect and the full cubic model, respectively, when the mixture contains q components. Recently, Panda^[17] obtained saturated R-optimal designs for quadratic canonical polynomial for $q$ mixture components as well as for different forms of cubic polynomial mixture models, that is, full cubic, cubic without a three-way effect and special cubic mixture models in three mixture components.

Zhu and Hao^[18] investigated A-optimal designs for the special cubic mixture model. Panda^[9] found A-optimal designs for the cubic model without a three-way effect for $q$ -mixture components. The work on finding optimal designs for Scheffé’s canonical polynomial model is, at most, limited to the cubic model in the current literature, and to the best of the author’s knowledge, the problem of finding optimal designs for Scheffè’s quartic polynomial model is not yet discussed in the literature. In this article, we have attempted to find saturated D-optimal designs for the quartic canonical polynomial model when the mixture contains three and four mixture components. The advantage of using a saturated design lies in its ability to estimate the parameters of interest with the minimum number of experimental runs, while also simplifying the computational procedure.

The article is structured as follows. Section 2 discusses the D-optimal design and the corresponding equivalence theorem. In Section 3, we obtain D-optimal designs for the Scheffè’s quartic polynomial model. Finally, we conclude the article with some discussion and conclusions in Section 3.

2. D-optimal Design and Equivalence Theorem

Let us consider a regression model of the form:

η (x) = f^{'} (x) β, x \in ζ,

(2.1)

where $η (x)$ denotes the expected response, $x$ is the input variable, and $f (x)$ is the regression function. Further, we shall assume that all responses are independent of each other and have constant variance. A continuous design $ξ \in Δ$ has the following form (see Kiefer ^[19]):

ξ = \{\begin{matrix} x_{(1)} & x_{(2)} & \dots \\ α_{1} & α_{2} & \dots \end{matrix} \begin{matrix} x_{(m)} \\ α_{m} \end{matrix}\}, x_{(j)} \in ζ, 0 < α_{j} < 1, \sum_{j = 1}^{m} α_{j} = 1,

where $x_{(1)},$ $x_{(2)}, \dots, x_{(m)}$ are the different design points defined over $ζ$ . $α_{j}$ is the weight associated with the point $x_{(j)}$ , j = 1, 2, …, m. Here $Δ$ is defined as the set of all continuous designs. A non-singular information matrix for a design $ξ \in Δ$ can be defined as:

M (ξ) = \sum_{j = 1}^{m} α_{j} f (x_{(j)}) f^{'} (x_{(j)})

(2.2)

over $ζ$ .

Throughout this article, we consider the design points of the form: $x^{'} = (r_{1}, r_{2}, \dots, r_{q}) \in ζ, r_{1}, r_{2}, \dots, r_{q} \geq 0$ and $r_{1} + r_{2} + \dots + r_{q} > 0$ , by $x ~ (r_{1}, r_{2}, \dots, r_{q})$ we shall mean that $x_{1}, x_{2}, \dots, x_{q}$ are in the ratio $r_{1} : r_{2} : \dots : r_{q},$ and by $x \leftrightarrow (r_{1}, r_{2}, \dots, r_{q})$ we shall mean that $x ~ (r_{1}^{*}, r_{2}^{*}, \dots, r_{q}^{*})$ for some permutation $r_{1}^{*}, r_{2}^{*}, \dots, r_{q}^{*}$ of $r_{1}, r_{2}, \dots, r_{q}$ . For further details, refer to the article of Chan and Guan.^[20]

Next, we define the D-optimal design and the equivalence theorem for the D-optimality criterion, which would be useful to derive the main result.

Definition 2.1. A design $ξ^{*} \in Δ$ with an information matrix $M (ξ)$ for model Equation (2.1) is called a D-optimal design if it minimizes $Det (M^{- 1} (ξ))$ over the set $ζ$ .

Kiefer and Wolfowitz^[21] proposed an equivalence theorem, stated as Theorem 2.1, that is quite useful for examining the necessary and sufficient condition for the determination of D-optimal design over the simplex region $ζ$ .

Theorem 2.1. A continuous design $ξ^{*} \in Δ$ is said to be a D-optimal design for the model Equation (2.1) if and only if

{Max}_{x \in ζ} d (x, ξ^{*}) = p

(2.3)

where $d (x, ξ) = f^{'} (x) M^{- 1} (ξ) f (x)$ and $p$ is the number of parameters involved in the model Equation (2.1). Furthermore, the supremum exists at the support point of $ξ^{*}$ .

Initial candidate design: A saturated design having $n$ number of design points is D-optimal if the weight assigned to each design point is 1/ $n$ (see Silvey ^[22]). Based on this result, we consider the initial candidate design (as mentioned in Table 1) for finding the D-optimal design for the quartic polynomial model for the three-component mixture.

Table 1.

Initial Candidate Design for D-optimal Design for Quartic Polynomial Model ( $q$ = 3).

Support Points			Weight Assigned to Each Support Point
$x_{1}$	$x_{2}$	$x_{3}$
1	0	0	1/15
0	1	0	1/15
0	0	1	1/15
½	½	0	1/15
½	0	½	1/15
0	½	½	1/15
$a$	$1 - a$	0	1/15
$a$	0	$1 - a$	1/15
0	$a$	$1 - a$	1/15
$1 - a$	$a$	0	1/15
$1 - a$	0	$a$	1/15
0	$1 - a$	$a$	1/15
$1 - 2 b$	$b$	$b$	1/15
$a$	$1 - 2 b$	$b$	1/15
$b$	$b$	$1 - 2 b$	1/15

3. D-optimal Designs for Quartic Model

The quartic model can be expressed as follows:

\begin{array}{c} η_{1} (x) = f_{1}^{'} (x) β_{1} = \sum_{i = 1}^{3} β_{i} x_{i} + \sum_{i < j}^{3} β_{i j} x_{i} x_{j} + \sum_{i < j = 1}^{3} γ_{i j} x_{i} x_{j} (x_{i} - x_{j}) + \sum_{i < j = 1}^{3} δ_{i j} x_{i} x_{j} {(x_{i} - x_{j})}^{2} \\ + β_{1123} x_{1}^{2} x_{2} x_{3} + β_{1223} x_{1} x_{2}^{2} x_{3} + β_{1233} x_{1} x_{2} x_{3}^{2} \end{array}

(3.1)

when $q = 3$ (see Gorman and Hinman)^[23]. In Equation (3.1), $f_{1} (x)$ and $β_{1}$ are the column vectors of dimension $15 \times 1$ each and are denoted by:

f_{1} (x) = {(\begin{array}{l} x_{1}, x_{2}, x_{3}, x_{1} x_{2}, x_{1} x_{3}, x_{2} x_{3}, x_{1} x_{2} (x_{1} - x_{2}), x_{1} x_{3} (x_{1} - x_{3}), \\ x_{2} x_{3} (x_{2} - x_{3}), x_{1} x_{2} {(x_{1} - x_{2})}^{2}, x_{1} x_{3} {(x_{1} - x_{3})}^{2}, \\ x_{2} x_{3} {(x_{2} - x_{3})}^{2}, x_{1}^{2} x_{2} x_{3}, x_{1} x_{2}^{2} x_{3}, x_{1} x_{2} x_{3}^{2} \end{array})}^{'}

β_{1} = {(β_{1}, β_{2}, β_{3}, β_{12}, β_{13}, β_{23}, γ_{12}, γ_{13}, γ_{23}, δ_{12}, δ_{13}, δ_{23}, β_{1123}, β_{1223}, β_{1233})}^{'} .

When $q \geq 4$ , the quartic model can be represented as (see Gorman and Hinman)^[23]:

\begin{array}{c} η_{2} (x) = f_{2}^{'} (x) β_{2} = Σ_{i = 1}^{q} β_{i} x_{i} + {ΣΣ}_{i < j}^{q} β_{i j} x_{i} x_{j} + {ΣΣ}_{i < j}^{q} γ_{i j} x_{i} x_{j} (x_{i} - x_{j}) \\ + {ΣΣ}_{i < j}^{q} δ_{i j} x_{i} x_{j} {(x_{i} - x_{j})}^{2} + {ΣΣΣ}_{i < j < k}^{q} β_{i i j k} x_{i}^{2} x_{j} x_{k} + {ΣΣΣ}_{i < j < k}^{q} β_{i j j k} x_{i} x_{j}^{2} x_{k} \\ + {ΣΣΣ}_{i < j < k}^{q} β_{i j k k} x_{i} x_{j} x_{k}^{2} + {ΣΣΣΣ}_{i < j < k < l}^{q} β_{i j k l} x_{i} x_{j} x_{k} x_{l} \end{array}

(3.2)

Here $f_{2} (x)$ and $β_{2}$ are column vectors of length $(\begin{matrix} q + 3 \\ 4 \end{matrix}) \times 1$ and are denoted by:

f_{2} (x) = {(\begin{array}{l} x_{1}, x_{2}, \dots, x_{q}, x_{1} x_{2}, x_{1} x_{3}, \dots, x_{q - 1} x_{q}, x_{1} x_{2} (x_{1} - x_{2}), x_{1} x_{3} (x_{1} - x_{3}), \dots, x_{q - 1} x_{q} (x_{q - 1} - x_{q}), \\ x_{1} x_{2} {(x_{1} - x_{2})}^{2}, x_{1} x_{3} {(x_{1} - x_{3})}^{2}, \dots, x_{q - 1} x_{q} {(x_{q - 1} - x_{q})}^{2}, x_{1}^{2} x_{2} x_{3}, \dots, x_{q - 2}^{2} x_{q - 1} x_{q}, x_{1} x_{2}^{2} x_{3}, \\ \dots, x_{q - 2} x_{q - 1}^{2} x_{q}, x_{1} x_{2} x_{3}^{2}, \dots, x_{q - 2} x_{q - 1} x_{q}^{2}, x_{1} x_{2} x_{3} x_{4}, \dots, x_{q - 3} x_{q - 2} x_{q - 1} x_{q} \end{array})}^{'}

β_{2} = (β_{1}, β_{2}, \dots, β_{q}, β_{12}, β_{13}, \dots, β_{(q - 1) q}, γ_{12}, γ_{13}, \dots, γ_{(q - 1) q}, δ_{12}, δ_{13}, \dots δ_{(q - 1) q}, β_{1123}, \dots, β_{(q - 2) (q - 2) (q - 1) q}, β_{1223}, \dots, β_{(q - 2) (q - 1) (q - 1) q}, β_{1233}, \dots, {β_{(q - 2) (q - 1) (q - 1) q})}^{'} .

In the next theorem, we obtain the D-optimal design $ξ_{1}$ for the model Equation (3.1).

Theorem 3.1. The design $ξ_{1}$ that allocates weight $\frac{1}{15}$ to each of the support points $x \leftrightarrow (1, 0, 0)$ , $x \leftrightarrow (\frac{1}{2}, \frac{1}{2}, 0)$ , $x \leftrightarrow (a, 1 - a, 0)$ and $x \leftrightarrow (1 - 2 b, b, b)$ with $a = \frac{1}{14} (7 - \sqrt{21})$ , $b = \frac{1}{22} (7 - \sqrt{5})$ is the D-optimal design for the model Equation (3.1).

Proof. Let $ξ_{1}$ be a saturated design that allocates weight $\frac{1}{15}$ to each of the support points $x \leftrightarrow (1, 0, 0)$ , $x \leftrightarrow (\frac{1}{2}, \frac{1}{2}, 0)$ , $x \leftrightarrow (a, 1 - a, 0)$ and $x \leftrightarrow (1 - 2 b, b, b)$ . For the design $ξ_{1}$ , the information matrix of the form Equation (2.2) is given by:

M (ξ_{1}) = [\begin{array}{ccccc} α_{1} U_{1} + α_{2} U_{2} & α_{3} V_{1} + α_{4} V_{2} & α_{5} X_{1}^{'} & α_{6} V_{1} + α_{7} V_{2} & α_{8} U_{1} + α_{9} U_{2} \\ α_{3} V_{1} + α_{4} V_{2} & α_{10} U_{1} + α_{11} U_{2} & α_{12} X_{2}^{'} & α_{13} U_{1} + α_{14} U_{2} & α_{15} V_{1} + α_{16} V_{2} \\ α_{5} X_{1} & α_{12} X_{2} & α_{13} U_{1} + α_{17} X_{5} = & α_{18} X_{3}^{'} & α_{17} X_{4}^{'} \\ α_{6} V_{1} + α_{7} V_{2} & α_{13} U_{1} + α_{14} U_{2} & α_{18} X_{3} & α_{19} U_{1} + α_{20} U_{2} & α_{21} V_{1} + α_{22} V_{2} \\ α_{8} U_{1} + α_{9} U_{2} & α_{15} V_{1} + α_{16} V_{2} & α_{17} X_{4} & α_{21} V_{1} + α_{22} V_{2} & α_{23} U_{1} + α_{24} U_{2} \end{array}]

(3.3)

where the submatrices are given by:

U_{1} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]; U_{2} = [\begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}]; V_{1} = [\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{matrix}]; V_{2} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}];

X_{1} = [\begin{array}{ccc} - 1 & 1 & 1 \\ - 1 & 0 & 1 \\ 0 & - 1 & 1 \end{array}]; X_{2} = [\begin{array}{ccc} 0 & - 1 & 1 \\ - 1 & 0 & 1 \\ - 1 & 1 & 0 \end{array}]; X_{3} = [\begin{array}{ccc} 0 & 1 & - 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}]; X_{4} = [\begin{array}{ccc} 1 & 1 & 0 \\ - 1 & 0 & 1 \\ 0 & - 1 & - 1 \end{array}]; X_{5} = [\begin{array}{ccc} 0 & 1 & - 1 \\ 1 & 0 & 1 \\ - 1 & 1 & 0 \end{array}]

The involved coefficients in Equation (3.3) are given by:

α_{1} = \frac{1}{10} + \frac{1}{15} (2 + 4 a (a - 1)) + \frac{1}{15} (6 b^{2} - 4 b + 1),

α_{2} = \frac{1}{60} - \frac{2}{15} a (a - 1) + \frac{1}{15} (2 - 3 b) b,

α_{3} = \frac{1}{120} - \frac{1}{15} (a - 1) a + \frac{1}{15} b (1 + 3 b (b - 1)),

α_{4} = \frac{1}{15} (1 - 2 b) b^{2},

α_{5} = \frac{1}{15} {(1 - 2 a)}^{2} (a - 1) a + \frac{1}{15} b {(1 - 3 b)}^{2} (2 b - 1),

α_{6} = - \frac{1}{15} {(1 - 2 a)}^{2} (a - 1) a + \frac{1}{15} {(1 - 3 b)}^{2} (b - 1) b (2 b - 1),

α_{7} = - \frac{2}{15} {(1 - 3 b)}^{2} b^{2} (2 b - 1),

α_{8} = \frac{1}{15} b^{2} (1 - 2 b (3 + b (6 b - 7))),

α_{9} = \frac{1}{15} b^{3} (2 + b (6 b - 7)),

α_{10} = \frac{1}{240} + \frac{2}{15} a^{2} {(a - 1)}^{2} + \frac{1}{15} b^{2} (2 + (9 b - 8) b),

α_{11} = \frac{1}{15} (1 - 2 b) b^{2},

α_{12} = \frac{1}{15} {(1 - 3 b)}^{2} b^{2} (2 b - 1),

α_{13} = \frac{2}{15} {(2 a^{3} - 3 a^{2} + a)}^{2} + \frac{2}{15} b^{2} {(b (6 b - 5) + 1)}^{2},

α_{14} = \frac{1}{15} {(1 - 3 b)}^{2} (b - 1) b^{2} (2 b - 1),

α_{15} = \frac{1}{15} b^{3} (1 + b (9 b - 6 b^{2} - 5)),

α_{16} = \frac{1}{15} {(1 - 2 b)}^{2} b^{4},

α_{17} = \frac{1}{15} b^{2} {(1 + b (6 b - 5))}^{2},

α_{18} = \frac{1}{15} b^{2} {(1 - 2 b)}^{2} {(1 - 3 b)}^{3},

α_{19} = \frac{2}{15} {(1 - 2 a)}^{4} {(a - 1)}^{2} a^{2} + \frac{2}{15} {(1 - 3 b)}^{4} {(1 - 2 b)}^{2} b^{2},

α_{20} = \frac{1}{15} b^{2} {(1 - 2 b)}^{2} {(1 - 3 b)}^{4},

α_{21} = - \frac{1}{15} (b - 1) b^{3} {(1 + b (6 b - 5))}^{2},

α_{22} = \frac{2}{15} b^{4} {(1 + b (6 b - 5))}^{2},

α_{23} = \frac{1}{15} {(1 - 2 b)}^{2} b^{4} (6 b^{2} - 4 b + 1),

α_{24} = - \frac{1}{15} {(1 - 2 b)}^{2} b^{5} (3 b - 2) .

Next, the function $D e t (M (ξ))$ is obtained as:

ψ (ξ) = D e t (M (ξ)) = c_{1} {(a - 1)}^{12} a^{12} {(2 a - 1)}^{18} b^{12} {(2 b - 1)}^{6} {(3 b - 1)}^{6}

(3.4)

Here, $c_{1}$ denotes a constant that remains unaffected by any design-related issues. The non-zero restriction of $D e t (M (ξ))$ in Equation (3.4) leads to the following:

0 < a < \frac{1}{2}, and 0 < b < 1 / 3.

(3.5)

To maximize Equation (3.4), we derive the partial derivatives of it with respect to $a$ and $b$ and set them equal to 0. Thus, we get:

\frac{\partial}{\partial a} ψ (ξ) = c_{2} {(a - 1)}^{11} a^{11} {(2 a - 1)}^{17} (1 + 7 a (a - 1)) b^{12} {(1 - 3 b)}^{4} {(1 - 2 b)}^{6} b^{12} = 0,

(3.6)

\frac{\partial}{\partial b} ψ (ξ) = c_{3} {(1 - 2 a)}^{18} {(a - 1)}^{12} a^{12} b^{11} {(2 b - 1)}^{5} {(3 b - 1)}^{3} (1 + b (11 b - 7)) = 0.

(3.7)

Similarly, in Equations (3.6) and (3.7), both $c_{2}$ and $c_{3}$ are constants that remain independent of any design-related factors. Now, solving the systems of Equations (3.6) and (3.7), we have the following cases:

(a) $a$ = 1, $a$ = ½;

(b) $b$ = 1/2, $b$ = 1/3;

(d) $1 + b (11 b - 7) = 0$ .

Out of the above cases, cases (a) and (b) are not feasible due to the restrictions mentioned in Equation (3.5). Further, cases (c) and (d) lead to the values $a = \frac{1}{14} (7 \pm \sqrt{21})$ and $b = \frac{1}{22} (7 \pm \sqrt{5})$ . Due to the constraints given by Equation (3.5), the values $a = \frac{1}{14} (7 + \sqrt{21})$ and $b = \frac{1}{22} (7 + \sqrt{5})$ are infeasible. Therefore, we consider the remaining feasible solutions of the system of Equations (3.6) and (3.7), which are as follows: $a = \frac{1}{14} (7 - \sqrt{21})$ and $b = \frac{1}{22} (7 - \sqrt{5})$ .

Replacing the values of $a = \frac{1}{14} (7 - \sqrt{21})$ and $b = \frac{1}{22} (7 - \sqrt{5})$ in the left hand side of Equation (2.3), we get the expression of $d (x, ξ_{1})$ . Due to its considerable length, the expression for $d (x, ξ_{1})$ has been omitted from this study; nevertheless, it can be provided by the authors upon request.

Next, using the ‘fminsearch’ command of MATLAB software, we find that ${Max}_{x \in ζ} d (x, ξ_{1}) = 15.$ This establishes the necessary and sufficient condition of the equivalence theorem. This proves the theorem.

Remark 2.1. We extend the findings presented in Theorem 3.1 to the case of four-component mixture experiments. Due to the high-dimensional structure of the associated information matrix and the complexity of the function $d (x, ξ),$ the formal proof of this result is not included here. However, the proof is available from the authors upon request. In this extended setting, the D-optimal design consists of 35 distinct support points. These include the points $x \leftrightarrow (1, 0, 0, 0)$ , $x \leftrightarrow (\frac{1}{2}, \frac{1}{2}, 0, 0)$ , $x \leftrightarrow (a, 1 - a, 0, 0)$ , $x \leftrightarrow (1 - 2 b, b, b, 0)$ and $x \leftrightarrow (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})$ where the constants $a$ and $b$ are defined as in Theorem 3.1.

Remark 2.2. The computation of the inverse of the information matrix becomes a bulky task because the information matrices are of large dimensions, for example, it shall be a $70 \times 70$ square matrix, when $q = 5.$ Also, the information matrix for model Equation (3.2) seems to be an unstructured one. Thus, we restrict ourselves to handling the problem up to $q \leq 4$ . However, we firmly believe that the above-mentioned results can be generalized to the q-mixture components. Therefore, we conjecture the following result for Scheffè’s quartic canonical polynomial model given in the form of Table 2, when the mixture contains $q$ components.

Table 2.

D-optimal Designs for Quartic Polynomial Model ( $q \geq 4$ ).

	Support Points	Number of Support Points	Weight Allocated to Each Support Point
$a = \frac{1}{14} (7 - \sqrt{21})$ $b = \frac{1}{22} (7 - \sqrt{5})$	$x \leftrightarrow (1, 0, 0, \dots, 0)$	$q$	1/ $λ$
	$x \leftrightarrow (1 / 2, 1 / 2, 0, \dots, 0)$	$(\begin{matrix} q \\ 2 \end{matrix})$	1/ $λ$
	$x \leftrightarrow (a, 1 - a, 0, \dots, 0)$	3 $(\begin{matrix} q \\ 2 \end{matrix})$	1/ $λ$
	$x \leftrightarrow (1 - 2 b, b, b, 0, \dots 0)$	3 $(\begin{matrix} q \\ 3 \end{matrix})$	1/ $λ$
	$x \leftrightarrow (1 / 4, 1 / 4, 1 / 4, 1 / 4, 0, \dots, 0)$	$(\begin{matrix} q \\ 4 \end{matrix})$	1/ $λ$
	Total	$λ = (\begin{matrix} q + 3 \\ 4 \end{matrix})$

4. Discussion and Conclusions

The present article obtains saturated D-optimal designs for Scheffè’s quartic polynomial model. To the best of our knowledge, this article is the first of its kind that obtains the optimal designs for the quartic canonical polynomial model.

The consideration of finding A-optimal, V-optimal and R-optimal designs for a quartic polynomial model can be interesting for future studies.

Footnotes

Acknowledgement

The authors would like to express their thanks to the referees and the Editor for their valuable suggestions that significantly improved the quality and readability of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Mahesh Kumar Panda

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D-optimal Designs for Quartic Polynomial Model for Mixture Experiment with Three and Four Ingredients

Abstract

Keywords

1. Introduction

Initial Candidate Design for D-optimal Design for Quartic Polynomial Model ( q = 3).

D-optimal Designs for Quartic Polynomial Model ( q ≥ 4 ).

Footnotes

Acknowledgement

Declaration of Conflicting Interests

Funding

ORCID iD

References

Initial Candidate Design for D-optimal Design for Quartic Polynomial Model ( $q$ = 3).

D-optimal Designs for Quartic Polynomial Model ( $q \geq 4$ ).