Abstract
Sudoku squares have been widely used to design an experiment where each treatment occurs exactly once in each row, column or sub-block. For some experiments, the size of row (or column or sub-block) may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Sudoku squares design (BISSD) is proposed. A general method for constructing BISSD is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Sudoku designs. The relative efficiencies of a delete-one-transversal balance incomplete Latin Square (BILS) design with respect to Sudoku design are derived. In addition, linear model, numerical examples and procedure for the analysis of data for BISSD are proposed
Highlights
A Sudoku design is a block matrix filled with different Latin letters or numbers, such that each letter or number appears in a row only once
It was presented by Subramani and Ponnuswamy (2009) for the construction of Sudoku designs of order k m2 only and proposed linear models for analyzing the data obtained from their design, and applied the designs to agricultural experiments
Researchers address a classical problem of experimental designs, in the construction of balanced incomplete Latin squares (BILS) design
Summary
A Sudoku design is a block matrix filled with different Latin letters or numbers, such that each letter or number appears in a row (column or sub-block) only once. Mahdian and Mahmoodian (2015) report that a balanced incomplete Sudoku design is an incomplete filled block boxes with some empty cells which satisfied that each letter appears only once in (row, column or sub-block) This balanced incomplete Sudoku design has been shown to be non deterministic polynomial time complete for the particular case of sub-blocks square. This paper proposed construction and analysis of balanced incomplete Sudoku square designs that is to select certain cells from a complete Sudoku square such that the remaining cells satisfy the condition of balanced occurrence of numbers. The method of constructing balanced incomplete Sudoku square designs is presented in section 2 with other related results and numerical examples. A natural way of constructing a BISSD is to select certain cells from a complete Latin square such that the remaining cells satisfy the condition of balanced occurrence of symbols It can be done by removing one or more “transversal”. J takes vales in (1, 2, . . . , k) and l is the symbol in the (i, j)-th cell of the BILS(k, r), μ, is the overall mean, αiis the ith row effect,βj is the jth column effect, τl is the effect of the lth treatment, and the errors eijlare the independent N(0, σ2)
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