A block design setting is an integer triple ecifying the numbers of treatments , blocks , and experimental units per block , respectively , available for an experiment . If , i.e . if it is not feasible to have a single replication of every treatment in each block , then is called a incomplete block design setting . In every 0cm 0cm 0pt; TEXT-ALIGN: justify" The BIB designs , when they exist , are known to be A , D , E , and many other optimalities and are universally optimal . Therefore , we usually attempt to find a BIB design for constructing the best design in the 0cm 0cm 0pt; TEXT-ALIGN: justify" Generally , a BIB design cannot exist unless two integers $ r $ and $ \lambda $ exist that satisfy two relationships and . Existence of these integers are necessary condition for the existence of a BIB design in the block design settings . Evenmore , there are different incomplete block design settings that we can not construct a BIB design for them . One of the most important cases , is irregular balanced incomplete block (IBIB) design settings . The relationships between parameter values of IBIB design settings are considered as a necessary condition for existence of BIB designs but no BIB design exists for them So that , it is difficult to find the best design in a 0cm 0cm 0pt; TEXT-ALIGN: justify" The base of this thesis is IBIB design settings and evenmore attempt to find a design as the best design for them . Since optimality criteria are different and usually have different results , the best design is a design that is known to enjoy many optimalities . In this thesis we study some of optimality criteria that have been studied in the previous research . Also , we submit new results for E_{4}- and MV-optimal designs . We attempt to constructe the best possible design for each of these criteria . The MV-optimal designs are very difficult to find for many given ettings . We also consider the optimality results for searching MV-optimal designs with a given setting . According to acquiring results by simulation and programming with R software , E_{4}-optimal design is A - and D-optimal design for IBIB design settings but it is impossible to be MV-optimal . Also , it is possible to find a MV-better than the other optimal designs .