The Challenge of Generating Spatially Balanced Scientific Experiment Designs

The development of the theory and construction of combinatorial designs originated with the work of Euler on Latin squares. A Latin square on symbols is an matrix ( is the order of the Latin square), in which each symbol occurs precisely once in each row and in each column. Several interesting research questions posed by Euler with respect to Latin squares, namely regarding orthogonality properties, were only solved in 1959 [3]. Many other questions concerning Latin squares constructions still remain open today. From the perspective of the Constraint Programing (CP), Artificial Intelligence (AI), and Operations Research (OR) communities, combinatorial design problems are interesting since they possess rich structural properties that are also observed in real-world applications such as scheduling, timetabling, and error correcting codes. Thus, the area of combinatorial designs has been a good source of challenge problems for these research communities. In fact, the study of combinatorial design problem instances has pushed the development of new search methods both in terms of systematic and stochastic procedures. For example, the question of the existence and non-existence of certain quasigroups (Latin squares) with intricate mathematical properties gives rise to some of the most challenging search problems in the context of automated theorem proving [16]. So-called general purpose model generation programs, used to prove theorems in finite domains, or to produce counterexamples to false conjectures, have been used to solve numerous previously open problems about the existence of Latin squares with specific mathematical properties. Considerable progress has also been made in the understanding of symmetry breaking procedures using benchmark problems based on combinatorial designs [5, 6, 9, 13]. More recently, the study of search procedures on benchmarks based on Latin squares has led to the discovery of the non-standard probability distributions that characterize complete (randomized) backtrack search methods, so-called heavy-tailed distributions [8]. In this paper we study search procedures for the generation of spatially balanced Latin squares. This problem arises in the design of scientific experiments. For example, in agronomic field experiments, one has to test and compare different soil treatments. Two different soil treatments may correspond to two different fertilizers or two different ways of preparing the soil. Most agronomic field experiments are implemented through randomized complete block designs (RCBD) where each block has as many experimental units as treatments [4]. Use of blocks is in most cases justified by spatial variability in fields, and this layout is an attractive way to organize replications. This approach to experimental design uses random allocation of treatments to plots, which is used to ensure that a treatment is not continually favored or handicapped in successive replications by user bias or some extraneous source of variation [2]. Although this