Construction of Weighted Second Order Rotatable Simplex Designs (Wrsd)

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. A rotatable simplex design is one of the new designs that have been suggested for fitting second-order response surface models. In this article, we present a method for constructing weighted second order rotatable simplex designs (WRSD) which are more efficient than the ordinary rotatable simplex designs (RSD). Using moment matrices based on the Simplex and Factorial Designs, and the General Equivalence Theorem (GET) for Dand Aoptimality, weighted rotatable simplex designs (WRSDs) were obtained. Aand Doptimality criterion was then used to establish the efficiency of the designs.


Introduction
A rotatable simplex design is one of the newly introduced designs in response surface experiment. Rotatable Simplex Designs have been suggested to have a very wide usage e.g in Food science, Business performance, Health sciences, Bioprocessing, Engineering, Construction Industry and so on as its performance was illustrated in Response Surface Analysis (RSA) of percentage crude oil removed by three factors. Rotatable Simplex Designs (RSDs) are constructed using the properties of a Simplex -lattice design (SLD), through Full Factorial Designs (FFDs) Design points from SLD are used to generate the original design points for the RSD. The levels of the SLD were increased by taking all the combination levels of the original points from the SLD such that the sum of all odd moments is zero. These points were then augmented with all the combination levels of the distance from the centre point ( ). Equation (1) was then solved to attain rotatability.
where the summation in the above relations is over the design points = 1, 2, … , .
The introduction of weights to RSD in this paper is for the purpose of improving the design by making it more efficient

Optimality Criteria and Efficiencies
Optimal designs are experimental designs that are generated based on a particular optimality criterion and are generally optimal only for a specific statistical model. The ultimate purpose of any optimality criterion is to measure the largeness of a non-negative definite × information matrix C.
The optimality criterion used in this study were from the family of matrix means ∅ = −1, 0 , introduced by Kiefer (1964) and is discussed in detail by Pukelsheim (1993).
Efficiency tests the goodness of a design.

Weighted Rotatable Simplex Designs
A WRSD is modified from the general RSD by separating it into Simplex and "Radius" Factorial blocks having weights ! and assigned to the Simplex block "# ! $ and 'Radius' factorial block "# $.
In this study, the rotatable WRSD will be expressed as: An outline of the procedure to be used in obtaining the WRSDs is as follows:

)*, *+ Optimal Designs
Optimal designs are experimental designs that are generated based on a particular optimality criterion and are generally optimal only for a specific statistical model. Here we have used the general equivalence theorem to compute the values of the masses assigned to the design points of the RSD to obtain D-optimal and A-optimal WRSDs. Optimality measures the largeness of a non-negative definite × matrix [ where C is the subset of M. In this study, the matrix [ will be the information matrix based on the full parameter system of the model.

)Ÿ, *+ Optimal Designs
Similarly, the general equivalence theorem is used to compute the values of the masses assigned to the design points of the RSD to obtain D-optimal and A-optimal WRSDs

D -Optimal Wrsd
D -optimal WRSD design is: Using the GET to obtain values of ! and which would give a D-optimal design. i.e solving ! and to satisfy

Efficiencies of the Designs
The performance of the RSD and WRSD was measured using the D-and A-criterion.

D -Efficiency
The performance of the WRSD in comparison to the RSD is measured by the D-efficiency which is defined by ± . Using the D-optimal ∅ 6 "[ y "&$$ values for the two designs from Table 1, the D -Efficiency values are: From the efficiencies, it is noted that the WRSD is 15.2% more D -efficient for two factors and 76% more D -efficient for three factors.

A -Efficiency
Similarly the A -optimal ∅ N! "[ y "&$$ values for the two designs from Table 1   From the efficiencies, it is also noted that the WRSD is 98.53% more A -efficient for two factors and 99.07% more A -efficient for three factors.

Conclusion
In this study, we have presented a method for constructing a WRSD. The constructed design has achieved estimation efficiency as shown by the results in relation to their moment matrices. These designs have also proved to be D-and Aoptimal.