Application of Optimization Techniques for Construction of Incomplete Block Designs

Abstract

Incomplete block designs are useful to conduct experiments when block sizes are to be kept low to maintain the homogeneity within blocks. Experimenters are often faced with the selection of a suitable design given number of treatments, number of blocks and block size(s). Several theoretical and computer aided methods of obtaining incomplete block designs are available in literature. In this project, we have dealt with the problem of obtaining incomplete block designs through the application of optimisation approaches. First Chapter gives the introduction, review of literature related to incomplete block design constructions, motivation and objectives of the project. In Chapter II we have proposed a constraint satisfaction approach to construction of incomplete block design with specified concurrence matrix. Construction of different classes of binary incomplete block designs viz. balanced incomplete block designs, regular graph designs, semi-regular graph designs etc., using the proposed approach are illustrated with examples. In the third Chapter we have presented a multi-step linear integer programming approach to construct a proper binary incomplete block design with specified parameters and concurrence matrix. Construction of different classes of binary incomplete block designs viz. balanced incomplete block designs, regular graph designs, semi-regular graph designs etc. using the proposed approach are illustrated with examples. Modification of the algorithm for obtain incomplete block designs for tests vs control(s) comparisons has also been shown and illustrated with examples. A catalogue of efficient incomplete block designs for 3 <= v <= 20,b >= v, 2 <= k <= min(10,v -1) with vb <=1000 is presented in Chapter IV. Here v denotes the number of treatments, b denotes the number of blocks and k denotes the block size To construct efficient incomplete block designs for given number of treatments, number of blocks and block size, we have utilized the idea that efficient incomplete block designs will be nearly balanced in nature. A sketch of obtaining concurrence matrix with nearly balanced structure has also been given. The proposed algorithm has been utilized to construct balanced treatment incomplete block designs for 2 <= v <=12,v <= b <= 50, 2 <= k <= v -1 where v denotes the number of test treatments, b denotes the number of blocks and k denotes the block size. A list of obtained designs in the above range is also presented. The layouts of the designs are available on Design Resources Server at http://iasri.res.in/design/. All the proposed methods have been implemented using R and SAS. An R package called ‘ibd’ has been developed and is available on cran.r-project.org/web/packages/ibd/index.html. SAS macros have been prepared and are available with the authors.