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Generating and Improving Orthogonal Designs by Using Mixed Integer Programming

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Generating and Improving Orthogonal Designs by Using Mixed Integer Programming European Journal of Operational Research 215 (2011) 629–638 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Stochastics and Statistics Generating and improving orthogonal designs by using mixed integer programming a, b a a Hélcio Vieira Jr. ⇑, Susan Sanchez , Karl Heinz Kienitz , Mischel Carmen Neyra Belderrain a Technological Institute of Aeronautics, Praça Marechal Eduardo Gomes, 50, 12228-900, São José dos Campos, Brazil b Naval Postgraduate School, 1 University Circle, Monterey, CA, USA article info abstract Article history: Analysts faced with conducting experiments involving quantitative factors have a variety of potential Received 22 November 2010 designs in their portfolio. However, in many experimental settings involving discrete-valued factors (par- Accepted 4 July 2011 ticularly if the factors do not all have the same number of levels), none of these designs are suitable. Available online 13 July 2011 In this paper, we present a mixed integer programming (MIP) method that is suitable for constructing orthogonal designs, or improving existing orthogonal arrays, for experiments involving quantitative fac- Keywords: tors with limited numbers of levels of interest. Our formulation makes use of a novel linearization of the Orthogonal design creation correlation calculation. Design of experiments The orthogonal designs we construct do not satisfy the definition of an orthogonal array, so we do not Statistics advocate their use for qualitative factors. However, they do allow analysts to study, without sacrificing balance or orthogonality, a greater number of quantitative factors than it is possible to do with orthog- onal arrays which have the same number of runs. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction 1999, p. 23). Orthogonality is important because without this prop- erty it is impossible to model the effect of one factor independently One of the few tools used in almost all branches of science is of the other factors. The orthogonality definition given above guar- experimental testing. Areas as diverse as medicine, chemistry, psy- antees that even qualitative factors can be analyzed. chology, engineering and biology have been using experiments to The study of orthogonal arrays was introduced by Rao (1945) in develop and prove their theories. Although experimental design the 1940s. The main techniques used to construct orthogonal ar- originated with the need for experiments in agriculture, it has been rays are: (1) Galois fields and finite geometries; (2) Codes; (3) Dif- widely used in many other domains, such as manufacturing and ference Schemes; (4) Hadamard Matrices; (5) orthogonal Latin industrial applications, clinical trials, military test and evaluation squares and F-Squares. For a review of orthogonal design creation, (T&E), and simulation. Hedayat et al. (1999, p. 298) credit we refer the reader to Hedayat et al. (1999). The study of how to approaches from Taguchi, in the main, and Deming, as leading to improve or create designs is still an active research topic, as can the ‘‘explosive increase in the use of design for product quality be seen, for example, in Lejeune (2003), Kleijnen (2005), van Beers improvement and monitoring in industrial processes...’’. and Kleijnen (2008) and Hernandez et al. (2011). Taguchi’s most important contribution to quality improvement In this paper, we limit the scope of our studies to problems in techniques is his emphasis on investigating both mean perfor- which all the factors are quantitative (either continuous or dis- mance and performance variation (Koch, 2002, p. 2). To this end, crete). If an array has the property that the maximum absolute Taguchi created several orthogonal arrays. These designs, which pairwise correlation between any two columns equals zero, we have been widely used and are incorporated in several statistical will refer to this as an orthogonal design. Note that orthogonal software packages, are described in more detail in Section 3. arrays (under the classical definition of Dey and Mukerjee (1999) Orthogonal arrays have several definitions according to the area seen above) are, by construction, orthogonal designs—but there of application. A common definition is the following: ‘‘An orthogo- are orthogonal designs that are not orthogonal arrays. nal array OA (N,n,m mn,g), having N rows, n(P2) columns, Analysts faced with conducting experiments involving quanti- 1 ÂÁÁÁ m m (P2) symbols and strength g( n), is a N n array, tative factors have a variety of potential designs in their portfolio. 1 ÂÁÁÁ n 6  with elements in the ith column from a set of mi distinct symbols This includes orthogonal arrays; two-level designs such as factorial (1 6 i 6 n), in which all possible combinations of symbols appear and fractional factorial designs, or Plackett–Burman designs; equally often as rows in every N g subarray’’ (Dey and Mukerjee, central composite designs; and space-filling designs like Latin  hypercubes. One reason for promoting two-level designs is their efficiency in terms of minimizing the variances of the estimated Corresponding author. Tel.: +55 12 39473671. ⇑ E-mail addresses: [email protected] (H. Vieira), [email protected] effects. Central composite designs that extend two-level designs (S. Sanchez), [email protected] (K.H. Kienitz), [email protected] (M.C.N. Belderrain). leverage this efficiency, as well as allowing quadratic effects to 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.07.005 630 H. Vieira Jr. et al. / European Journal of Operational Research 215 (2011) 629–638 be explored. Space-filling designs are advantageous when little is the construction method works best with fewer groups of discrete known about the nature of the response surface, and when factor factors. combinations in the interior of the region of exploration are of po- Our approach does allow direct construction of mixed designs. tential interest. We generalize Hernandez’s continuous-factor MIP formulation to However, in many experimental settings involving discrete-val- deal with mixed designs, and to assure the balance and space-filling ued factors, none of these designs are suitable. Designs for the spe- properties that good designs should have. A design is said to be bal- cific combination of factor levels may not be readily available— anced if, for each factor, the number of objects in each of its levels is particularly if the factors do not all have the same number of levels. equal. The balance property is important from a mathematical per- The orthogonal arrays defined and constructed as above may re- spective because it allows correct analysis of heteroscedastic exper- quire more runs than the allowed budget, in part because the fac- iments (see Bathke, 2004). Balance is also desirable from a practical tor levels may mandate a large number of runs (estimating main perspective on several counts, particularly for exploratory investi- effects requires c 1 degrees of freedom for a qualitative factor gations over broad factor ranges where linearity and constant var- À with c levels, while a single degree of freedom suffices for a quan- iance assumptions are often inappropriate. First, it is rare for an titative factor). At the same time, focusing on two-level designs (or analyst to know a priori the relationship between the factors and even central composite designs) may be undesirable because they the response variability if they do not already know the relationship do not sample at all levels of interest for each factor, and the space- between the factors and the expected response, so constructing a filling designs in the literature are intended for continuous-valued design that accommodates known heteroscedastic behavior is gen- factors or factors with the same number of levels. Rounding the erally not an option. By using a balanced design, an analyst need not specified levels to accommodate discrete-valued factors can ad- make assumptions about the nature of the response variability. Sec- versely affect the orthogonality properties of the design (Sanchez ond, exploratory studies often involve more than one potential re- and Wan, 2009). A flexible method for constructing efficient sponse, and it may not be apparent ahead of time which one(s) orthogonal designs is needed. will wind up being of most interest. For example, in a disaster relief In this paper, we present a mixed integer programming (MIP) scenario, the primary response might be the total amount of food method that is suitable for constructing orthogonal designs. We and medical supplies distributed, and the factors might represent also show cases where our approach can improve existing orthog- various delivery resource levels and protocols. However, if the onal arrays. Our formulation generalizes and extends that of Her- alternatives are similar in terms of the total goods distributed, the nandez (2008), and makes use of a novel linearization of the analyst might then be interested in examining results from the correlation calculation. We describe our approach in Section 2, same design for other responses, such as the time or cost associated and present some numerical results in Section 3. In Section 4 we with the disaster relief efforts. Third, balance makes it easier for an summarize our findings and briefly discuss the benefits of this analyst to focus on specific region(s) of the design space that appear approach. interesting. For example, local analyses of regions where trade-offs between cost and effectiveness can be examined in detail may be more informative than global analyses that show effectiveness 2. Our approach tends to improve as resource levels increase. Balance and space-filling behavior are both useful design char- We base our approach on the work of Hernandez (2008) and acteristics when there is an interest in the interior of the region of Hernandez et al. (2011), who use MIP to create nearly orthogonal exploration or when little is known about the nature of the re- Latin Hypercubes with any number of design points n, provided sponse surface.
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