Some problems in block designs and optimality criteria

Abstract

Jacroux (1985) extended the definition of Regular graph (RG) designs of Mitchell and John (1976) to Semi-regular graph (SRG) designs, and studied the type 1 optimality of block designs. In this investigation, the construction and optimality of some more SRG designs, their dual designs and RG designs with unequal block sizes are carried out, which are not discussed by Mitchell and John (1976). Moreover, it is established that a class of two associate partially efficiency balanced (PEB) design is, in fact RG or SRG designs. RG designs are also obtained by adding two RG designs having the same treatments and same size of blocks. Further, it is shown that the existence of a BIB design with _ = 2 implies the existence of a RG design. Again, the adjacency graph of RG designs is given, and thereby some new RG designs are obtained. Also, two series of Group Divisible (GD) designs from v _ 3 (mod 6) BIB designs are constructed, and established that one series is Eand#8722;optimal. Further, Semi-Regular Group Divisible (SRGD) designs are obtained, by adding two SRGD designs having the same treatments and same size of blocks. Also, it is shown that (S-) type Balanced Treatment Incomplete Block (BTIB) designs for comparing test treatments with a control are partially efficiency balanced (PEB) designs. Again, it is established that (R-) type binary BTIB and GDT designs are simple PEB designs, and binary BTIB designs with _o = _1 are proper EB designs.