www.ccsenet.org/jmr Journal of Mathematics Research Vol. 2, No. 4; November 2010 Construction of Optimal Odd Fractional Designs in Mixed Series Experiment Using Semi Latin Square Williams, E. E. (Corresponding author) Department of Maths, Stats and Computer Science, University of Calabar, Calabar, Nigeria P.M.B. 1115, Calabar, Nigeria Tel: 234-803-547-0457 E-mail: edemwilliam@yahoo.com Akpan, S. S. Department of Maths, Stats and Computer Science, University of Calabar, Calabar, Nigeria P.M.B. 1115, Calabar, Nigeria. Tel: 234-803-380-8695 Ugbe, T. A. Department of Maths, Stats and Computer Science University of Calabar, Calabar, Nigeria Nduka, E. C. Department of Mathematics & Statistics, University of Port Harcourt Port Harcourt, Nigeria Abstract A new technique for the construction of optimal odd fractional designs in mixed series experiment is developed. The construction begins by classifying all the experimental points in the experimental region into groups where the levels of each factor is centred at zero, thereafter the P point trial designs is formed by picking the support points from the first group to the last depending on the size of the designs using some optimality checks. The technique obtains a P-point optimal design using loss of information as an optimality Criteria. Numerical illustrations confirm the effectiveness of this technique. Keywords: Latin Square, Mixed Experiment, Optimality Criteria, Optimality Check, Odd-Fractional Designs 1. Introduction There are many experiments designed to study the effects of variables largely under the control of the experimenter on other variables which are functions of the controlled variables (Raghavarao, 1971). The controlled variables are called factors and the functions are the response variables. The experiment is conducted by changing the settings of the factors and observing the changes which result in the response variables. The settings of a factor for which observations are made on the response variables are called the levels of the factors. (Pazman, 1987) A complete factorial experimental design requires that observations be made on the response variables for all combinations of the levels of the factors. These combinations will be referred to as treatment combinations. A fractional factorial experimental design requires that only a fraction of these treatment combinations be studied. (Meyer, 1971) Catalogs of fractional factorial designs for the case when all factors have two levels, and three levels are mostly not available. In symmetrical factorial experiment, we can confound higher order interactions without losing any information on main effects. In mixed factorials, pm x qn we can confound r independent effects and interactions of factors at p levels and s independent effects and interactions of factors at q levels in blocks of size pm-rqn-s. White and Hulquist (1995) has given methods for the designs and analysis of confounding plans for the pm x qn factorial experiment, p and q primes, using a technique of combining elements from distinct finite fields. An equivalent theoretical base utilizing ideal theory was obtained by Raktae (1999) and Banjere (1970) showed that it will be possible to provide the confounding plans for mixed factorials of the types 5n x 8m, where n and m are any two positive integers. The aim of this paper is to present a methodology which will yield optimal odd fractional designs in mixed series experiment using semi-Latin squares. Published by Canadian Center of Science and Education 19