The optimal design theory has been developed under various assumption like independence and normality errors. The optimal design theory assumes absence from disturbances like missing observations, outlying observations or inadequacy of assumed model, etc. These assumptions may, however, be violated in real life; thus rendering even an optimal design poor. In order to overcome such a situation, we have to think of designs which are insensitive or robust against such disturbances. A design is said to be robust against one or more of above disturbances if it remains insensitive to presence of one or more of the above disturbances in terms of design properties. In the field of experimentation, once the experiment has been laid in the field, the observation can be destroyed or lost during the course of experimentation. Missing data can cause serious problem and may render even a well-planned experiment useless. This is one of the main reason for the popularity of this area of research over last two decades. However, there are several experiments where the loss of a whole block is more likely than the loss of an individual observation. The consequences of disconnected block design may be so sever that the original aims of experiment will be spoiled; pair wise treatment contrast will not be estimable and it will not be possible to test the usual null hypothesi that all treatments have the same effect. Although the design may be robust in the sense of connectedness against the type of disturbances described above, the residual design obtained may not be efficient as compared to the original design. The efficiency criterion is therefore based on the efficiency of the residual design. According to this criterion, a design is robust against loss of observations if the efficiency of residual design is close to the efficiency of original design. Researchers have investigates methods for guarding against a disconnected design because of observation loss during the experiment and there are various criteria for studding robustness of designs, One of these criterion of robustness is connectedness and other is efficiency criterion which are introduced in chapter two. In chapter three, some conditions for an arbitrary binary block design to be maximally robust against the unavailability of data are expressed in terms of design parameters. These conditions do not take full account of the information given by basic design parameters, thus further stronger conditions are also introduced in this chapter. In chapter four and five, the optimality and relative efficiency of eventual designs which are obtained after the loss of fixed number of blocks when the planned design is a BIBD and binary variance balanced block design are considered .