Mixture Designs Generated by Orthogonal Arrays Developed Using Difference Schemes

This paper presents an algorithm for constructing mixture designs based on orthogonal arrays developed using difference schemes. The algorithm can also be applied to constrained mixture experiments. The algorithm allows the generation of efficient mixture designs for Scheffé canonical polynomials. KeywordsMixture experiments, Difference schemes, Restricted region, G-efficiency.


Introduction
In experiments with mixtures, the response is a function only of the proportions of the components present in the mixture and not of the total amount of the mixture. If is the proportion of ℎ component, = 1, 2, … , , then 0 ≤ ≤ 1, ∑ =1 = 1. (1) These restrictions force the factor space, of the components to take form of a ( − 1) dimensional simplex. The general purpose of mixture experimentation is to estimate the properties of an entire multicomponent system from only a limited number of observations. These observations are taken at preselected combinations of the components to determine which of the combinations in some sense optimize the response.
In many practical situations, one can encounter certain additional constraints that are placed on some or all component proportions besides (1). These are of the form 0 ≤ ≤ ≤ ≤ 1; = 1, 2, … , .
where, and denote the lower bound and upper bound for the component proportion ; = 1, 2, … , . These supplementary restrictions limit the experimentation to some sub-region of the simplex, thereby altering the shape of the experimental region from a simplex to an irregularly shaped convex polyhedron inside the simplex. In such situations, directing the design and modelling only to the sub-region can help in lowering the experimentation cost and time and increasing the precision of model estimates.
Restricted region has been explored in different ways. Extreme vertices designs (McLean and Anderson, 1966) were developed to satisfy both the constraints (1) and (2). A partial solution to the restricted exploration problem was provided by identifying the point of maximum interest and considering an ellipsoidal region around it (Thompson and Myers, 1968). Subsets of the extreme vertices were shown to provide precise estimates of the parameters of a linear model (Snee and Marquardt, 1974). An improved algorithm for mixture designs with components bounded by upper and lower bounds was established (Nigam et al., 1983). Computers were used to develop designs in constrained mixture spaces for the quadratic model (Snee, 1975). The restricted mixture region was investigated using symmetric-simplex design (Saxena and Nigam, 1977). A method of construction of mixture designs for the examination of the restricted region using factorials was deliberated (Murthy and Murty, 1983). The construction of mixture designs using Taguchi's fixed element orthogonal arrays was discussed (Singh et al., 2018).
Difference schemes were defined as a tool for constructing orthogonal arrays of strength two (Bose and Bush, 1952). An example of a difference scheme of index two based on the additive group of Galois Field was stated (Bose and Bush, 1952). The construction of difference scheme of index two, (2 , 2 , ), where is a prime or a power of a prime was studied (Masuyama, 1969;Jungnickel, 1979;Xu, 1979;Xiang, 1983). The methods of recursive construction of orthogonal arrays based on a series of difference schemes were specified (Shrikhande, 1964;Mukhopadhyay, 1981).
In this paper, we present an algorithm for constructing mixture designs based on orthogonal arrays generated using difference schemes. We have examined and compared the designs constructed through this algorithm with the existing designs based on G-efficiency. The manageable number of design points help in reducing the cost and time in statistical experiments.

Orthogonal Arrays Based on Difference Schemes
Hypercubus of strength ′ ′ were defined (Rao, 1946). Later, the definition of hypercube of strength was extended to cover a wider class of arrays called orthogonal arrays (Rao, 1947). An × array with entries from is said to be an orthogonal array ( , , , ) with levels, strength and index (for some in the range 0 ≤ ≤ ) if every × sub-array of contains each -tuple based on exactly times as a row. If = 1, then such arrays are referred to as orthogonal arrays of index unity (Bush, 1952). Orthogonal arrays can be constructed using difference schemes.
An × array with the entries from a finite abelian group, is called a difference scheme based on binary operation +, represented by ( , +), if it has the property that for all and with 1 ≤ , ≤ , ≠ , the vector difference between the th and th columns contains every element of equally often.

Design Criteria
If denotes the × design matrix, then a useful criterion for evaluating the design is the minimum-maximum variance criterion. This refers to minimizing the maximum variance of prediction over the experimental region, where the prediction variance at the point (1 × row vector) is given by 2 and = ( ′ ) −1 ′. Computation of the maximum variance provides a criterion of how close is a design to optimality. The G-efficiency or Global efficiency of the design is given by, where, is the number of model parameters, is the number of design points and is the maximum value of over the experimental region.
As a practical rule of thumb, any design with a − ≥ 50% could be called good for practical purposes (Wheeler, 1972).
We now present Orthogonal Array Based Mixture Design (OABMD) algorithm for constructing component mixture designs using orthogonal arrays based on difference schemes.
Step 2: Create a matrix of order × which is symmetrical but not orthogonal, having all elements as integers with sum of elements in each row and each column being zero. The choice of is arbitrary and is useful in getting more vertices of the experimental region in the design (Murthy and Murty, 1983).
Step 3: Identify the minimum value in each column of × and subtract it from all the entries of that corresponding column to create a new matrix .
Step 4: Obtain the row totals for matrix . Divide the entries of each row of by its corresponding row total to obtain a new matrix . The resultant matrix is a mixture design satisfying 0 ≤ ≤ 1 and ∑ =1 = 1, being the proportion for ℎ component.
Further, if the mixture experiment has to be performed in the restricted region, where each component is bounded by lower or upper bounds, or both, then proceed as follows: Step 5: Rank the components in order of their increasing ranges = ( − ) such that 1 has the smallest range and has the largest range, assuming range to be inversely proportional to the importance (in terms of cost, effectiveness, etc.) of the components in the experiment.
Step 7: In case lies beyond the specified bounds, generate candidate design points. There may be multiple candidate points corresponding to a given design point. The candidate points are generated by adjusting the level of one of the components by a quantity equal to the difference between the substituted upper or lower bound and the computed value for . Additional points are produced only from those components whose adjusted levels remain within the limits of the components.
We have illustrated the OABMD algorithm for generating mixture designs using orthogonal arrays based on difference schemes. These designs have been found to be efficient designs.
Multiply it with a symmetric and non-orthogonal matrix, = − , where and denote identity matrix and a matrix of all ones respectively both of order . In this case, = 3.
Other choices of with corresponding G-efficiency values of and for fitting Scheffé's canonical polynomials are listed in Table 1.  1,0,0), to the resulting 9-run design matrix, which initially was giving a G-efficiency of 27.63% for fitting (5).
Remarkably, it is detected that the G-efficiency of the unconstrained mixture design constructed using the orthogonal array with any choice of (as listed in Table 1) is 75% for fitting Scheffé's linear model and 80% for fitting Scheffé's quadratic model. The same result holds for the constrained three component mixture experiment (Snee and Marquardt, 1974;Saxena and Nigam, 1977).
Using a symmetric and non-orthogonal matrix, * = − , where and denote identity matrix and a matrix of all ones respectively both of order . In this case, This design matrix consists of three repeat runs and has a G-efficiency of 80%, 83.33% and 70.88% for fitting (3), (4) and (5) respectively.
Using the OABMD algorithm, the constrained mixture design, * , is * = ( This design has thirteen distinct runs and a G-efficiency of 80%, 83.33% and 84.70% for fitting (3), (4) and (5) respectively as compared to the fifteen design runs considered by McLean and Anderson (1966) yielding a G-efficiency of 57.16%, 71.24% and 93.33% for fitting (3), (4) and (5) respectively. Considering only the non-repeat runs in both the design matrices, * and * , yields a G-efficiency of 94.11% and 76.92% for fitting (3) and (4) respectively. Other choices of * with corresponding G-efficiency values of * and * for fitting Scheffé's canonical polynomials are listed in Table 2. Interestingly, it is observed that the G-efficiency of the unconstrained mixture design constructed using the orthogonal array * with any choice of * (as listed in
Using a symmetric and non-orthogonal matrix, * * = ( + 1) − , where and denote identity matrix and a matrix of all ones respectively both of order + 1. In this case, = 4. .
This design matrix consists of all distinct runs and has a G-efficiency of 73.96% and 93.75% for fitting (3) and (4) respectively.
Using the OABMD algorithm, the design matrix, 1 is as follows: This design matrix also consists of all distinct runs and has a G-efficiency of 73.96% and 93.75% for fitting (3) and (4) respectively. On the contrary, the design matrix suggested by Saxena and Nigam (1977) had twenty design points having a G-efficiency of 49.78% and 76.62% for fitting (3) and (4) respectively. Other choices of * * with corresponding G-efficiency values of unconstrained and constrained mixture design for fitting Scheffé's canonical polynomials are listed in Table 3. Table 3. G-efficiency of unconstrained and constrained mixture designs corresponding to different choices of * * for fitting Scheffé's canonical polynomials Remarkably, it is detected that the G-efficiency of the unconstrained and constrained mixture design, the set of constraints as given by Saxena and Nigam (1977), constructed using any choice of * * (as listed in Table 3) is 73.96% for fitting (3) and 93.75% for fitting (4).
This design matrix consists of all distinct runs and has a G-efficiency of 84.48% for fitting Scheffé's linear model.
Other choices of * * * with corresponding G-efficiency values of unconstrained mixture design for fitting Scheffé's linear model are listed in Table 4.  It can be observed from the above table that the G-efficiency for fitting (3) is 84.48% for the unconstrained design matrix attained by using the orthogonal array with any choice of the * * * matrix (as listed in Table 4). However, in case fitting of higher order models (4) and (5) is required, the boundary points, mid-point of the design runs or the extreme vertices may be added to the existing design matrix for fitting the models and subsequently the computation of G-efficiency.

Difference Scheme ( , , ) of Strength
A difference scheme ( , , ) of strength can be used to construct an ( , , , ). If the difference scheme itself is already an orthogonal array of strength − 1 or if it can be written as the juxtaposition of difference schemes −1 ( / , , ), then it can be used to construct an ( , + 1, , ).
Using a symmetric and non-orthogonal matrix, * * * * = ( + 1) − , where and denote identity matrix and a matrix of all ones respectively both of order ( + 1). In this case, = 7. .
This design matrix consists a repeat run and has a G-efficiency of 88.88% for fitting Scheffé's linear model. Nevertheless, considering only the distinct runs of the design matrix 3 yields a Gefficiency of 94.11% for fitting (3).
Other choices of * * * * with corresponding G-efficiency values of unconstrained mixture design for fitting Scheffé's linear model are listed in Table 5.  Interestingly, the same value of G-efficiency for fitting (3) is achieved for the unconstrained design matrix obtained by using the orthogonal array with any choice of the * * * * matrix (as listed in Table 5). However, in case fitting of higher order models (4) and (5) is required, the boundary points, mid-point of the design runs or the extreme vertices may be added to the existing design matrix for fitting the models and subsequently the computation of G-efficiency.

Conclusion
The orthogonal arrays developed using difference schemes have been considered in our proposed OABMD algorithm. The designs, hence, constructed have manageable number of design points that help in reducing the cost and time in statistical experiments. Furthermore, the flexibility in choice of matrix allows for enhanced variety of design points. When the region of interest is predefined, the proposed OABMD algorithm can be customized to explore a restricted space.
The constructed designs have a sufficiently high G-efficiency that make them suitable for practical purposes.