The modified balanced confounded design algorithm for constructing n-point d-optimal global designs

Introduction

It has been established that of all the N-point possible designs for both Symmetric and Asymmetric response surface designs that could be obtained by the Balanced Confounded Designs Algorithm (BCDA) for constructing N-point D-optimal designs proposed by (Umoren and Isaac, 2010) [4] , the balanced confounded designs are indeed D-optimal. Thus, the search for D-optimal designs was to be restricted only to the class of balanced designs. However, we know that the aim of all iterative search procedures like the Variance Exchange Algorithm (Atkinson and Donev, 1992) [1] , the Steepest Ascent Method (Pazman, 1986) [2] , the Combinatorial Procedure (Onukogu and Iwundu, 2008) [3] and the most recent procedure, the Balanced Confounded Designs Algorithm (Umoren and Isaac, 2010) [4] , is to locate the global optimum (best design) within the shortest possible moves or iterations in order to save cost, time and computer space and not just to locate an optimal design. This means that though a design may be balanced and hence, D-optimal, yet it may not be the best design (global optimum) among its class of designs. The BCDA though proved to be superior in efficiency to the Combinatorial Procedure, yet, did not take cognisance of which group of support points that should be taken while searching for global optimum so that the global optimum could be located within the shortest possible moves or iterations. This work which is indeed a modification of the Balanced Confounded Designs Algorithm, will take care of this drawback in order to ensure that the number of moves or iterations required to reach the global optimum (best design) is further reduced to the barest minimum.

Symmetric and Asymmetric Designs

Symmetric design is the design in which the factors   , , ...... A B C N appear at the same level.

For example, This procedure is as follows;

 

Step1: Group the support points according to the number of minus and plus    and zeros   0 and number them as

Step 2: Partition each group

 

Step3: Choose N number of support points that would go into the class. Let j V be defined as a vector of points consisting of 11 n selected from 1 G , 22 n selected from 2 G and rr n selected from k G where 11 n is the number of support points from 1 G that will make the N number of support points and so on. For example, if N = 20, and we define a class of design as

(1) 20 V 7: 6: 6 :1  , this means that, take 7, 6, 6, and 1 support point from 1 2 3 ,, G G G and 4 G respectively.

 

Step4: Using combination mathematics, determine the number of designs from that class and label them as

Step5: Obtain the information matrix

for each of the design and compute its determinant as

 

 

Step7: Tabulate your result above as follows;

 

Step1: Group the support points according to the number of minus and plus    and zeros   0 and number them as

Step 2: Partition each group , 1, 2, 3...

where the sub-groups are being referred to as sub-design measures.

 

Step3: Choose N number of support points that would go into the class. Let j V be defined as a vector of points consisting of 11 n selected from 1 G , 22 n selected from 2 G and rr n selected from k G where 11 n is the number of support points from 1 G that will make the N number of support points and so on. For example, if N = 20, and we define a class of design as

(1) 20 V 7: 6: 6 :1  , this means that, take 7, 6, 6, and 1 support point from 1 2 3 ,, G G G and 4 G respectively.

 

Step4: Using combination mathematics, determine the number of designs from that class and label them as

without splitting the sub-design measures since they are already balanced.

 

Step5: Obtain the information matrix

for each of the design and compute its determinant as

and test each design whether it is balanced or unbalanced. A design is tested for balance by calculating their loss of information L.I (See Umoren and Isaac, 2010) [4]  

Step6: Tabulate your result as balanced and unbalanced designs.

 

Step7 Define another vector of points for N and carry out   Step4 and  

The table of this Algorithm is given thus;

Given the experimental area

The support points are grouped as follows;

1 S: Define a vector of points, j V such as

For the Balanced Confounded Design Algorithm, the possible designs will be 2 since the sub-design measures are not split and they are;

We obtain the information matrix (

) for each of the designs as follows;

( 5 ) The results from table 2.1 and 2.2 above showed that for the Combinatorial Procedure, it took 8 moves or iterations to locate the global optimum whereas, using the BCDA, only 4 moves were used in locating the global optimum. More so, the BCDA showed that the global optimum was among the balanced designs. It was therefore concluded that the BCDA was superior in efficiency when compared with the Combinatorial Procedure, thus, the search for global designs was to be restricted to the class of balanced design using the BCDA..

Exploration with the Balanced Confounded Design Algorithm

Since it has been established that the Balanced Confounded Design Algorithm (BCDA) is superior in efficiency to the Combinatorial Procedure, we thus work with the BCDA for both the symmetric and the asymmetric designs using the examples of Umoren and Isaac, 2010 [4] . to be the experimental model for the K 3 series, for 2 3 designs, the support points are grouped as;

Exploration with the

A tabulation of the result of the analysis for a 7 point, 2 3 response surface design is given as

to be the experimental region, the support points are grouped as;

A tabulation of the result of the analysis is given in the table below; x    to be the experimental region, the support points are grouped as;

Exploration with

A tabulation of the result for balanced and unbalanced designs for a 12-point designs for a 4 2 response surface design is given in the table below;

Proposed Modification

From Table 3.1, Table 3.2 and Table 3.3, remove 2 columns, that is, the columns for Unbalanced Designs and Max determinant for Unbalanced Designs respectively since the global design does not come from the unbalanced set.. In its place, add a column for Global Optimum, the result are summarized in the tables below; 3 response surface designs using the Balanced Confounded Design Algorithm, the Global optimum is found in the set of balanced designs in which all the support points from 1 G were taken. If we restrict the search for the global optimum to the class of designs in which all the support points from 1 G are taken, it will take only 2 iterations or moves as against 4 iterations to locate the global optimum. Again, from table 4.2, it could also be observed that in Constructing a 13-point D-Optimal Balanced Designs for 2 G . In this case, 2 moves are required to reach the global optimum, the same number of moves the Balanced Confounded Design Algorithm used.

Conclusion

We have shown that in constructing a 2 3 , 2 x 2 3 and afoot_0 2 Npoint D-optimal response surface designs using the Balanced Confounded Design Algorithm (BCDA), it will take fewer moves or iterations to locate the global optimum if we restrict the search for global optimum to the class of designs in which all or more support points from 1 G are taken as against the traditional Balanced Confounded Design Algorithm (BCDA). This algorithm is called the Modified Balanced Confounded Design Algorithm (MBCDA). The result of this work has shown that the Modified Balanced Confounded Design Algorithm (MBCDA) is superior in efficiency to the Balanced Confounded Design Algorithm (BCDA) because in Constructing a 7-point D-Optimal Designs for 2 3 response surface designs, by using the BCDA, it will take 4 moves to locate the global optimum whereas by using the MBCDA, it will take only 2 iterations or moves to locate the global optimum. Again, in Constructing a 13-point D-Optimal Balanced Designs for 2 x 2 3 response surface design, if the MBCDA is used, the global optimum will be reached after only 12 moves or iterations as against 30 moves if the Balanced Confounded Design Algorithm is used. Finally, in Constructing a 12-point D-Optimal Designs for

References

1. Atkinson AC, Donev AN. Optimal Experimental Designs; Oxford Science Publication, 1992.
2. Pazman A. Foundation of Optimal Experimental Design; D-Reidal Publishing Company, Boston, 1986.
3. Onukogu IB, Iwundu MP. A Combinatorial Procedure for Constructing D-Optimal Exact Designs; Statistic. 2008; 67(4): pp. 415-423.
4. Umoren MU, Isaac AA. On Constructing N-Point DOptimal Designs From Balanced Confounded Designs: ICASTOR Journal of Mathematical Sciences. 2010; 4(2):183-196.