supersaturated designs with odd number of runs

Addition of runs to a two-level supersaturated design

“…The extended SSD d $ is represented as an (n+r)  m matrix X $ ½X 0^A0 0 , where A is an r  m matrix with entries71 of the additional r runs. X $ attains the new lower bound to E(s 2 ), given in Theorems 2.1 and 2.2, when (i) A is an E(s 2 )-optimal SSD(r, m) using the bounds given by Das et al (2008) or Suen and Das (2010), depending upon whether r is even or odd, respectively, and (ii) AX 0 ¼ 0, for even m or AX 0 ¼ E for odd m, where 0 is a null matrix and E is a r  n matrix with entries 71.…”

“…For any two-level SSD(n, m) represented as X, a measure of non-orthogonality is defined as *-The designs also attains lower bound to E(s 2 ) given by Das et al (2008) or Suen and Das (2010) for the respective classes. #-The design is optimal according to Theorem 2.2 and has the value of E(s 2 ) smaller than the lower bound given by Das et al (2008).…”

“…Nguyen (1996) obtained a lower bound to E(s 2 ). Since then many improved lower bounds on E(s 2 ) have been obtained by several authors (see, for example, Nguyen and Cheng, 2010;Bulutoglu and Ryan, 2008;Das et al, 2008;Suen and Das, 2010). An SSD is optimal when it attains lower bound to E(s 2 ).…”

“…Recently SSDs for odd number of runs have also been studied extensively (see, for example, Nguyen and Cheng, 2008;Bulutoglu and Ryan, 2008;Suen and Das, 2010;Gupta et al, 2010). In an unbalanced design, the symbol À 1 (or + 1) appears an extra time in each column of the design.…”

“…But this is not a feasible solution because the experimenter has already run the experiment with n design points and r additional runs have to be added with the already experimented runs. The second possibility is to add r runs to the already used E(s 2 )-optimal SSD (n, m) and use the existing bounds given by Das et al (2008) and Suen and Das (2010) depending upon (n +r) is even or odd for evaluating the efficiency of the extended design. But this bound may be difficult to attain because the sub-design with n runs is already E(s 2 ) optimal using these bounds.…”