The   purpose   of   this   paper   is   to   developed   an   algorithm   and   a   computer   software   to   implement   the   work   of   Hedayat   and   Pesotan(2007).   2     The   Basic   Design   Principles   Underlying   the   Development   of   our   Algorithm   2.1   The   Linear   Model  

A  2 ^k  factorial  design  is  a  set  of  level  combinations,  i  =  (i 1 ,  i 2 ,  .  .  .  ,  i k ),  i  =  1,  2,  .  .  .  ,  2 ^k ,   where  i j  =  0  or  1,  j  =  1,  2,  .  .  .  ,  k.  Let  y i  be  the  response  corresponding  to  the  ith  combination   level.   Then   the   general   linear   model   containing   the   main   effects   and   one   two   -   factor   interaction   as   given   in   Hedayat   and   Pesotan   (2007)   is  

Where  

N   =   2 ^k :   =   number   of   level   combinations  

X ij   =   1,   if   factor   j   appears   in   the   ith   combination   level   X ij   =   -­‐1,   if   factor   j   do   not   appear   in   the   ith   combination   level   e i :   =   random   error   which   is   assumed   to   be   normal   with   mean   0   and   constant   variance.   2.2     The   g(k,   1)   -­‐   Design  

  In   Hedayat   and   Pesotan   (2007),   a   2 ^k   factorial   design   is   called   a   g(k,   1)   -   design   if   and   only   if:  

(i) It  is  capable  of  providing  an  unbiased  estimate  for  each  of  the  parameters  provided  by   the   linear   model   (1),  

(ii) It   is   saturated,   that   is,   it   contains   (k+2)   level   combinations   which   corresponds   to   the   number   of   parameters   in   the   model.  

A  nonsingular  square  matrix  X  of  order  (k+2)  whose  entries  are  -­‐1  and  1  is  called  a  g(k,  1)  -   matrix   if   and   only   if   it   is   in   the   form,  

                (2)  

where     w i   =   x ij x ir ,     i   =   1,   2,   .   .   .   ,   k+2;   the   sub-­‐matrix   X 1   is   called   the   core   of   X.     If   -­‐1   is   replaced   by   0   in   the   matrix   X,   a   g(k,   1)   -   design   is   produced.   Thus   constructing   a   g(k,   1)   -   design   is   equivalent   to   constructing   a   g(k,   1)   -   matrix.   2.3   The   D   -   Optimal   Design    

Let   Ω   be   a   class   of   all   n   -   square   matrices.   Then   an   n   -   square   matrix   M   is   said   to   be   D   -   optimal   in   Ω   if ,   for   any   other   n   -   square   matrix   Q   in   Ω.  

2.4   Index   of   Interacting   Columns   of   Core   Matrix   X 1  

Let   r   and   s   be   two   columns   of   X 1   which   interact   to   give   W.   The   product   w   =   r   o   s   is   called   a   shur   product   of   r   and   s.   Let   f ++   be   the   frequency   of   ++   pair   between   r   and   s,   f +-­‐   the   frequency   of   +   -­‐   pair   between   r   and   s,   f -­‐+   the   frequency   of   -­‐   +   pair   between   r   and   s,   and   f -­‐   -­‐   the   frequency   of   -­‐   -­‐   pair   between   r   and   s.   Then   the   index   of   r   and   s   is   defined   as           (3)  

The   following   lemma   is   in   Hedayat   and   Pesotan   (2007).   Lemma  

Let  G  (k,  1)  be  a  class  of  all  design  matrices  g(k,  1).  Let  T(r,  s)  be  a  design  matrix,  then   with   the   interacting   columns   r   and   s,   i(r,   s)   ≥   1.   3.   The   Design   of   Algorithm  

The   sequence   of   steps   for   our   algorithm   is   as   follows:   Step   0:  

Input  the  number  of  factors  k,  the  number  of  combination  level  N,  and  N  by  (k+2)  matrix  of  -­‐1   and   1   entries   X,     Step   1:   Compute    

Select   all   possible   N ^*   square   sub   -   matrices   A n   =   (1   B n   W n )   of   order   (k+2),   n   =   1,   2,   .   .   .   ,   N ^* ,   from   X,   where   w n   =   r n   o   s n ,   r n   and   s n   are   the   interacting   columns   of   B n .   Step   2:   Set   n:   =   1.   Is   n   ≤   N ^* ?  

If   Yes:   compute   det(A n )   and   go   to   step   3.   If   No:   Go   to   step   6   Step3:   Is   det(A n )   =   0?  

If   Yes:   A n   is   not   a   design   matrix.   Set   n:   =   n+1   and   return   to   step   2   If   No:   Go   to   step   4   Step   4:  

Compute   the   frequencies   f ++ ,   f +-­‐ ,   f -­‐+ ,   and   f -­‐   -­‐   of   the   interacting   columns   r   and   s.   Compute   the   index   i ⁿ (r,   s)   of   the   interacting   columns   r   and   s.  

Step   5:  

Is     i ⁿ (r,   s)   ≥   1?  

If   Yes   :   A n   is   a   design   matrix.   Set   n:   =   n+1   and   return   to   step   2   If   No   :   A n   is   not   a   design   matrix.   Set   n   :   =   n+1   and   return   to   step   2   Step   6:   Compute      

In   step   1,   N   =   1,   so   we   have   only   one   4   by   4   sub-­‐matrix   that   can   be   selected   from   X,   and   that   is   X  

In   step   2,   n   =   1   =   N ^* ,   so   we   compute   det(A 1 )   =   det(X)   =   2 ⁴ .  

In   step   3,   ,   so   we   go   to   step   4.  

In   step   4,   we   compute   the   index   of   the   interacting   columns,   column   1   and   column   2,   of   the   core   matrix   as   i(1,   2)   =   1.  

The   matrix   X   is   a   g(2,   1)-­‐matrix   since   i(1,   2)   =1   and     .  

The   unbiased   estimate   of   the   parameters     is   the   solution   of   the   following   system   of   linear   equations.  

That   is,    

We   need   a   5   by   5   sub-­‐matrix   A   of   X   to   find   the   unbiased   estimate   of   the   five   parameters .  There  are  56  such  sub-­‐matrices  of  which  36  are  g(3,  1)-­‐  matrices   each   with   determinant   2 ⁵   and   index   interacting   column   i(1,   2)   =   1    

The   matrix  

is   a   D   -   optimal   2 ⁵   design   matrix,   and   the   corresponding   unbiased   estimates   of   the   parameters  

That   is,  

5.     Conclusion  

We   have   studied   the   work   of   Hedeyat   and   Pesotan   (2007)   carefully,   and   then   developed   an   algorithm   to   implement   it.   The   algorithm   so   developed   is   coded   in   a   high   level   programming   language,   JAVA.   Our   algorithm   is   used   in   constructing   a   D-­‐optimal   g(2,1)   and   g(3,1)-­‐designs.   We   observed   that   a   D-­‐optimal   design   is   not   unique   for   k>2.   In   particular,   for   k   =   3   there   are   36   g(3,1)   D-­‐optimal   designs.  

The  unbiased  estimates  of  the  parameters  in  the  linear  model  for  the  two  designs  are  

obtained,   respectively.   Since   the   D   -   optimal   design   is   not   unique   for   k   >   2,   the   unbiased   estimates   of   the   parameters   varies   from   one   design   to   another.   One   should   therefore   seek   the   D   -   optimal   design   that   will   minimize   the   sum   of   squares   of   the   errors.   The  program  code  for  our  algorithm  is  available  for  the  interested  reader  by  contacting  any  of   the   authors.   REFERENCES  

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