-optimal and minimax-optimal cyclic supersaturated designs via multi-objective simulated annealing
Introduction
A two level design is called saturated when the number of factors (columns) m equals to , where n is the number of experimental runs (rows). A two level supersaturated design is a two level factorial design in which the number of experimental runs n is smaller or equal to the number of factors m. For each factor of a two level design there are two possible settings known as levels, which can be coded as . Any combination of the levels of all factors under consideration is called a treatment combination. Let [, be the design matrix of the experiment in which, each row represents an m treatment combination and each column gives the sequence of factor levels. For each factor, both level values are of equal interest. Thus we consider designs with the equal occurrence property, where all columns consist of elements equal to 1 and elements equal to , when n is even. The designs with the equal occurrence property are called balanced designs. The last row of the designs presented in this paper is a row of 's. If we omit this row, we obtain designs for odd number of rows, with the number of 's to be one less than the number of 's in each column. These designs are nearly-balanced designs.
A supersaturated design can save considerable cost when the number of factors is large and a small number of runs is desired or available. The usefulness of these designs relies upon the realism of effect sparsity, namely, that the number of dominant active factors is small. Therefore, for situations where there is no prior knowledge of the effects of factors, but a strong belief in factor sparsity, and the aim is to screen out active factors, experimenters should use supersaturated designs. For more details regarding the usage of supersaturated designs see Holcomb et al. (2006) and Gilmour (2006).
Section snippets
Optimality criteria for supersaturated designs
Orthogonality between all pairs of columns of the model matrix, which is formed from the design matrix by appending a column of 1's as the first column, is desirable for estimating main effects or factors. This condition cannot be satisfied for all pairs of columns in a supersaturated design where . Therefore, we try to construct designs as near orthogonal as possible. We present here three optimality criteria for constructing and evaluating supersaturated designs.
criterion: Let be
Cyclic structured supersaturated designs
To accommodate a large number of factors in a supersaturated design, Nguyen and Cheng (2006) proposed the use of cyclic balanced incomplete block designs (BIBDs). Their approach involved the creation of columns as cyclically generated blocks of a BIBD.
In our approach, we maintained the cyclic structure of generators although our construction is not based on BIBDs. Our goal is to construct supersaturated designs with n rows and columns, where n and q are even. In particular, each
Simulated annealing search for cyclic supersaturated designs
Single objective simulated annealing or simply simulated annealing (SA), is commonly said to be the oldest among metaheuristics and has its origins in statistical mechanics (Metropolis et al., 1958), and it was Kirkpatrick et al. (1983) and Cerny (1985) that translated it as a search algorithm for combinatorial optimization problems. We assume some basic familiarity with SA concepts. The concepts necessary for a description of the SA can be found in Laarhoven and Aarts (1987). In this section
Results
In this section, the -optimal and minimax-optimal cyclic structured supersaturated designs that we constructed with the aid of the multi-objective SA algorithm, given in the previous section, are presented. The balanced supersaturated designs given below have n runs and factors, where q is even. Furthermore, we present the corresponding near balanced supersaturated designs for runs and factors.
Table~1, Table~2 list the values of the optimality criteria of
Finding bounds for via bounds on and
In the following theorem we derive a lower bound on for certain values of . Furthermore, assuming -optimality we establish an explicit formula for . Theorem 4 Let be a balanced supersaturated design with n runs and factors. Let and . Then . Let and . Then . Let and . Then . In addition, if the design is -optimal (1) becomes equality for any positive
Acknowledgments
We would like to thank two anonymous referees for their detailed comments and suggestions. The research of the second author was financially supported by Greek State Scholarships Foundation (IKY). The research of the first and third author was financially supported from the General Secretariat of Research and Technology by a grant PENED 03ED740.
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