Statistics and Probability Letters 129 (2017) 269–274 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Upper bound on the number of multi-level columns in equally replicated optimal designs minimizing the E (f NOD ) criterion Vasilis Chasiotis a, *, Stratis Kounias b , Nikolaos Farmakis a a Department of Mathematics, Aristotle University, 54124, Thessaloniki, Greece, b Department of Mathematics, University of Athens, 15784, Zografou, Athens, Greece, article info Article history: Received 6 March 2017 Received in revised form 2 June 2017 Accepted 7 June 2017 Available online 23 June 2017 MSC 2010: 62K15 62K05 62K99 Keywords: Supersaturated designs Non-orthogonality Algorithm Optimality criteria abstract This article provides an upper bound on the number of multi-level columns, denoted by M, in equally replicated optimal designs, minimizing the E(f NOD ) criterion. These designs are called E(f NOD )-optimal. Applying an algorithm and by row juxtaposition, fourteen E(f NOD )-optimal designs are constructed. © 2017 Elsevier B.V. All rights reserved. 1. Introduction A few derived E (f NOD )-optimal designs may be supersaturated designs (SSDs). The SSDs are used to screen a few active effects in experiments when there is a large number of factors and a few experimental runs are available. SSDs are useful since the runs are usually expensive in order to avoid the elimination of factors by the absence of data. Booth and Cox (1962) proposed the E (s 2 ) criterion for a two-level design. Yamada and Lin (1999) defined the criteria ave χ 2 and max χ 2 , in order to evaluate a three-level SSD regarding the dependency between any two of its factors. Cheng and Tang (2001) derived upper bounds on the number of columns at two-level SSDs from coding theory and bounds on E (s 2 ) criterion. Fang et al. (2004) derived upper bound on the number of columns in SSDs for 2 ≤ N /q ≤ q and 1 ≤ r ≤ N /q, where N is the number of runs, q is the number of levels in each column and r is the maximum number that appear any of the q 2 level combinations in any two columns. The paper is organized as follows. In Section 2, the class of equally replicated designs is given and the notion of E (f NOD ) criterion is presented. In Section 3, minimizing the E (f NOD ) criterion proposed by Fang et al. (2003), an upper bound on the number of multi-level columns in equally replicated optimal designs is obtained. In Section 4, a table with the minimum values of E (f NOD ) criterion and the upper bounds on M for N ≤ 100 and 3 ≤ q ≤ 5 is presented. The construction methods of E (f NOD )-optimal designs and the constructed E (f NOD )-optimal designs are provided in Section 5. * Corresponding author. E-mail addresses: chasiotisv@math.auth.gr (V. Chasiotis), skounias@math.uoa.gr (S. Kounias), farmakis@math.auth.gr (N. Farmakis). http://dx.doi.org/10.1016/j.spl.2017.06.008 0167-7152/© 2017 Elsevier B.V. All rights reserved.