AN ALGEBRAIC AND COMBINATORIAL APPROACH TO THE CONSTRUCTION OF EXPERIMENTAL DESIGNS

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1 Geotec., Const. Mat. & Env., ISSN: (P), (O), Japan AN ALGEBRAIC AND COMBINATORIAL APPROACH TO THE CONSTRUCTION OF EXPERIMENTAL DESIGNS Julio Roero 1 and Scott H. Murray 2 1 Departent of Inforation Sciences and Engineering, University of Canberra, Australia; 2Departent of Matheatics and Statistics, University of Canberra, Australia. ABSTRACT: Experiental design is a well-known and broadly applied area of statistics. The expansion of this field to the areas of industrial processes and engineered systes has eant interest in an optial set of experiental tests. This is achieved through the use of cobinatorial and algebraic approaches. As such, the present study states the theoretical basis to construct and enuerate experiental designs using non-isoorphic atheatical structures in the for of atrix arrangeents called orthogonal arrays (OAs). These entities are characterized by their nuber of rows, coluns, entries (sybols), and strength. Thus, each different colun could represent soe easurable feature of interest (teperature, pressure, speed). The runs, expressed through OA rows, define the nuber of different cobinations of a particular design. Siilarly, the sybols allocated in OAs entries could be the distinct levels of the phenoenon under study. During the OA construction process, we used group theory to deal with perutation groups, and cobinatorics to create the actual OAs following a particular design. The enueration process involved the use of algebraic-based algoriths to list all possible cobinations of arrays according to their isoorphic equivalent. To test isoorphis, we used graph theory to convert the arrays into their corresponding canonical graph. The outcoes for this study are, firstly, a powerful coputational technique to construct OAs fro 8 to 80 runs; and secondly, additions in the published list of orbit sizes and nuber of non-isoorphic arrays given in [1] for 64, 72, and 80 runs. Keywords: Orthogonal arrays, Cobinatorics, Experiental design, Engineering paraeter design 1. INTRODUCTION The engineering or scientific ethod is the approach of solving probles through the efficient application of scientific principles based upon a well-structured theoretical knowledge [2]. In applying the aforeentioned approach, engineers undertake experients or tests as an intrinsic and natural part of their jobs. As such, sound statisticalbased experiental designs are extraordinarily useful within the engineering profession aiing to iprove process and systes characterized by suitable engineering specifications. These specifications are described as several controllable variables associated with the overall syste/process perforance. Therefore, knowing the ain factors and their interactions, the engineer is able to analyze, for instance, process yield, process variability, developent tie, and operational costs [3], [4]. Latest iplications in using experiental tests are related with engineering paraeter design. Within this discipline, it is possible not only to develop new products/processes, but also to enhance the existing ones. To cite soe exaples in paraeter design, we have the evaluation and coparison of basic design configurations; the appraisal of different alloys in strength of aterials; the selection of design paraeters to ake the product / process to work within a specified tolerance range under a wide variety of field conditions (we say the design is being robust); and the deterination of the key factor cobinations affecting a particular product perforance. Besides their flexibility of being easily ipleented in engineering design [2], our arrays are atheatically ade so we can perfor their cobinatorial enueration. This last idea is the one presented in this paper. Our interests in following a purely atheatical point of view in dealing with experiental designs are, firstly, the developent of algoriths which actually calculate the objects we are defining theoretically; secondly, to answer concrete questions in group theory to explore ore of it; and thirdly, to deal with coplexity theory such as to able to answer the proble whether two graphs, given by their adjacency atrix, are in fact, isoorphic [5]. We initially present the ain concepts of group theory which underlie the algoriths used to construct the arrays. We ove further with the tools provided by the theory of graphs, to ap an array into its equivalent canonical graph to be able to indirectly assess isoorphis between the arrays. We then present a pseudo code to ipleent a backtrack search algorith to look through the 66

2 different orbits which confor the orbit space of a particular array of design type t. 2. COMBINATORIAL SETTING Orthogonal arrays (OAs) are related to cobinatorics, finite fields, geoetry, and error correcting codes. Fig. 1 shows an exaple of an OA of strength two (transposed). of factor transforations and the corresponding autoorphis group. Furtherore, we used graph theory to represent the orthogonal array as a colored graph. This approach allowed us to define a canonical representative of an orbit of OAs using the canonical graph [9]. This leads to iprove the coputational tie and eory usage for the calculations. We also ipleented the backtrack search and the lexicographical least run algoriths to find the arrays and as a decision criteria to drop a leaf [1], [10]. 3.1 Algebraic Setting Fig. 1 Strength two OA (transposed) Each of the four possible rows does appear in the array, and they appear the sae nuber of ties (picked the in groups of tt = 2 coluns). This is the fundaental property that defines an OA [6]. In the previous exaple, only three sybols appeared, 0, 1, and 2. Thus, we say the array has 3 levels. Note that each level ay represent a particular cobination of physical properties: fuel with additive or not, 20 o C or 80 o C, achine running or stopped, and so forth. Following the sae exaple, we have 3 coluns each representing a variable under study (additive, teperature, achine s condition). In a siilar way, our array F has 12 rows or runs, each representing a particular cobination of the different variable levels. Matheatically, we represent an OA of strength tt, ss levels, NN runs, and aa variables as: OOOO(NN, SS ii aa jj, tt). For our particular array FF, we have OOOO(12, , 2) or equivalently, a design type UU(3 1, 2 2 ) [6], [7]. A strength-t OA assures that all possible cobinations of the levels in a variable of up to t occur together equally often [6]. Thus, the strength of the array defines the effects of each individual variable (factor) that can be taken in consideration when assessing the design [6], [8]. Also, according to the strength t required, we will be able to calculate soe interactions between factors as well. Such experients are called fractional factorial designs [7]. Our particular interest is in OAs of strength 3, which can odel all ain factors in a particular phenoenon under study and up to two interactions between the factors. 3. MATHEMATICAL BACKGROUND This section shows how we perfored the construction and enueration of strength three orthogonal arrays. We initially explain the atheatical approach used for the perutation coputational ipleentation. We applied group theory to define the full group Consider FF NNNN as the set of all eleents of the for: FF NNNN = FF: FF iiii aa NN dd, FF iiii SS jj (1) called the set of all fractional factorial designs (FFDs). We define the groups GG ρρ, GG γγ, and GG σσ, acting on FF NNNN as GG ρρ = SSSSSS(NN), the group of all perutations acting on rows; GG γγ = SSSSSS(DD), the group of perutations acting on coluns; and GG σσ = SSSSSS(SS), the perutation group acting on sybols. Thus, ρρ εε GG ρρ defines an action: FF ii,jj FF ρρ NNNN = FF ρρρρ,jj (2) siilarly, γγ GG γγ defines an action: kk SSSSSS(JJ kk ) SSSSSS(DD) FF γγ NNNN = FF ii,γγγγ (3) and σσ GG σσ defines an action: σσ = (σσ 1,, σσ) kk SSSSSS(JJ kk ) SSSSSS(SS) σσ σσ = FF jj ii,jj (4) FF NNNN Definition 1. Consider the set FF NNNN of NN dd arrays, and G the group of perutations. We define the action of the group G on the set FF NNNN as a ap φφ: GG FF NNNN FF NNNN such that: FF ee = FF, FF FF NNNN ; ee, identity eleent. (FF)(gg 1 gg 2 ) = (FFgg 1 )gg 2 gg 1, gg 2 GG. Corollary 2. Let the ap define a group hooorphis such that : GG 1 GG 2 where GG 1 and GG 2 are both perutation groups. Then is a one-to-one ap if and only if kkkkkk( ) = ee Proof. Assue the ap is one-to-one. It holds that (ee) = ee 2, the identity eleent of GG 2. Thus, ee is the only apped eleent into ee 2 by, so kkkkkk( ) = ee. Assue now that kkkkkk( ) = ee. For all gg GG the eleent apped into (gg) are the eleents of the right coset {eeee} = {gg} showing that 67

3 is one-to-one. Definition 3. We have the following appings: ρ : Sy(N) Sy(F ND ) (5) γγ : SSSSSS(JJ kk ) SSSSSS(FF NNNN ) (6) kk σ : Sy S j k Sy(F ND ) (7) It follows fro definitions 1 and 3, I ρ, I γ, I σ Sy(F ND ) (8) G = < IIII ρ, I γ, I σ > (9) G = G ρ G σ G γ (10) where G σ G γ = k=1 Sy S j,k Sy(J k ) or equivalently, G = ρ (ρ) σ (σ) γ (γ) Sy(F ND ) (11) Definition 4. Let G be a group of perutations acting on a non-epty set F ND ; we define the orbit of G containing F to the equivalence class given by: Orb G (F) = {Fg g G} (12) we define the orbit space as the faily of all equivalence classes obtained fro G acting on an eleent F of the set F ND. Let F and T both eleents of F ND. We say they are isoorphic if there exists a perutation g G such that F = Tg. The previous expression can also be written as F = T g. Suppose now that a group G acts on a set F ND ; then, for each eleent F F ND, we redefine the equivalence class (orbit) containing F as: O G (F) = {F g g G} (14) Lea 5. Suppose that a group G acts on a set F ND ; then, for each F F ND, G = stab(f) O g (F) (15) Proof. Suppose that g 1 and g 2 are in the right coset of stab(f) and g 1 = πg 2 for soe π stab(f). Thus, Fg 1 = (F)(πg 2 ) = (Fπ)g 2. On the other hand, suppose that Fg 1 = Fg 2, then F = Fg 2 g 1 1 iplying that g 2 g 1 1 stab(f); therefore g 1 and g 2 belong to the sae right coset of stab(f) if and only if Fg 1 = Fg 2. It follows that there is a bijection between the eleents in Orb G (F) and the right cosets of stab(f), thus G = stab(f) #of right cosets of stab(f) = G = stab(f) O g (F) and the result follows. Theore 6. (Burnside's Theore) Suppose the group G acting on a set F ND ; then, the nuber of orbits in F ND ; is given by 1 G stab(f) (16) F F ND Proof. Using lea 5, we have: 1 G stab(f) = 1 O g (F) (17) F F ND F F ND suppose there are O S orbits in F ND, and F in the orbit O g ; then, F F ND 1 O g (F) = O S. Definition 7. Let G be the group of all row, colun, and sybols perutations, and F F ND. The set of all isoorphiss fro F to F is called the autoorphis group of F and is denoted by Aut(F) The eleents of Aut(F) are called autoorphiss of F. Using our previous results fro Eqs. (15), (16) and (17); the length of the G-Orbit of G is the nuber of distinct objects isoorphic to it. Thus, Orb G (F) = G Aut(F) (18) 4. ENUMERATION USING GRAPH THEORY In the previous section, we explained how we atheatically set up the conditions for the construction and enueration of strength t orthogonal arrays. In this section, we forally explain their construction and how we perfor the enueration by apping the OAs as canonical graphs. 4.1 Mapping an OA to its Equivalent Canonical graph Let F ND the set of fractional factorial designs with F F ND. Let F ij be a particular entry in the array F, where i: i th row and j: j th colun. We define S ρ the set of all N-tuples which represent the different rows of the arrayf. Then, S ρ = {ρ 1,, ρ N } {ρ 1,, ρ N } and F ij S ; where S is the set 68

4 containing the different possible sybols for a particular design U. Definition 8. A colored graph is a triple F G = (V, E, Γ), where V is the set of the different vertices, v, of the graph; E is the set of edges, and Γ a ap fro V to a set of colors C. Definition 9. We define the neighbor of a vertex v x V as η(v x ) = v y V v x, v y E}. An isoorphis F G F G = (V, E, Γ) is a bijection η: V V such that: v x, v y E η(v x ), η v y E and Γ(v x ) = Γ v y Γ η(v x) = Γ η v xy v x, v x V. Proposition 10. Let F F ND. We construct a colored graph F G = (V, E, Γ) for the array F as follows: a) The set of the vertices V, is ade on the eleents ρ i, i: 1,2,, N; γ j, j: 1,2,, d and σ jx, j: 1, d; x S ; corresponding to the rows, coluns, and the distinct levels of the array. b) The set of the edges, E, ade on all of the E 1 = {ρ i, σ jfij } and E 2 = {γ j, σ jfij } i: 1,, N and j: 1,, d. c) The set of the edges, E, ade on all of the vertices ρ i with color C ρ ; all of the vertices γ with color C γ ; and all of the vertices σ jx with color C σx. Thus, F G has three partitions: rows, coluns, and levels. Matheatically, we express this as V = C ρ C γ C σx. Siilarly, we write the set of the edges as E = E 1 E 2 (C ρ C γ ) (C σx C γ ). It follows that the cardinalities V and E are given by d V = N + k i + d (19) I dd EE = dddd + kk ii II (20) To characterize the graph of an OA, we define the colun-color classes to the disjoint union of color classes C ρ, C γ, and C σx. The total nuber of colors of F G is 2 + and each row-vertex is adjacent to sv sybol-vertices. Moreover, each sybol-vertex is adjacent to exactly one colunvertex, where sv = C i i=1 Lea 11. Let SS FFFF the set of all coloured graphs, and the ap defined as : F ND SS FFFF that takes an array F F ND to the corresponding colored graph F G SS FFFF ; thus, is an injection. Proof. The nuber of vertices of F G does not depend upon F but only on the design type U and the run size N. To deterine row, sybol, and colun vertices, we have the following color partition proposition: Proposition 12. Let F F ND of strength t 1 and run-size N. Thus, a) F G has a tripartite partition (C ρ, C γ, C σx ) with C ρ = N, C γ = k=1 a k, and C σx = k=1 a k S k. b) Every vertex v V has a valence ν denoted by ν = V(v) c) For the colun-vertex set C σx, we denote the valency for c σx C σx as ν x, where ν x is the unique eleent {1,, }. d) Let s S, the set of different sybols (levels). Then c σx C σx such that {s, c σx } E, the set of edges for soe k in {1,, }. e) Fro d), it follows that the valence of a sybolvertex is given by ν(v) = N ν(c σx ) + 1 = N v x + 1 (21) f) Let ρ C ρ and γ C γ ; thus, there exists a path of length two fro ρ to γ through a vertex in V, and this path is unique. Definition 13. Let an orthogonal array be of design type U and run size N. Consider also a colored graph which satisfies all of the points given in 12 above. Thus, we call the colored graph as being of the type (U; N), which fors a sub-set of F G. Lea 14. Let F 2 and F 2 F ND OAs with designs type UU and NN. Let also F G1 ; F G2 S FG be their corresponding graphical representations. Thus, F 1 and F 2 are isoorphic arrays if and only if F G1 and F G2 are isoorphic graphs. Proof. Suppose F 1 and F 2 are isoorphic arrays F ND ; then F 1 = F g 2 ; g G. Because g G, it follows that g is a product of g ρ, g γ, and g σ perutations; and fro c) in proposition (10), we have F G1 = F g 1 = (F 2 ) = F G2. Therefore, F G1 and F G2 are isoorphic graphs. Suppose now that F G1 and F G2 are isoorphic graphs; thus, g G such that F G1 = F g G2. Because F G1 and F G2 satisfy all of the points given in proposition (10), they are tripartite and the perutation g preserves the graph coloring. Therefore, g can be written as the product of g ρ, g γ, and g σ perutations acting on F 1 and the coposed ap takes F 1 to F 2. 69

5 Exaple. Assue we have an orthogonal array with design type U(3 1, 2 2 ) and run size N = 6 (a fraction of the array shown in Fig. 1); then, the set of vertices is ade up on the eleents ρ i, i {1,,6}; and γ j, j {1,,7}; and σ jx, j {1,2,3}, x {1,2} (see proposition (10)). We assign one color to the set of rows C ρ ; one color to the partition C γ1 C γ corresponding to the level 3; one color to the second partition C γ2 C γ corresponding to the two levels of the two coluns with the sae sybol level (0,1); and one color for the set of sybols C σx. We have in total V = 6 + r i + 3 = 16 vertices, and E = edges. The colored graph of this array is shown in Fig i help us to save iportant coputational tie and eory-usage resources). Furtherore, the recently ade array is then apped to its equivalent canonical graph in order to be copared against the orbit-representative one. If it turns out that the two graphs are isoorphic, the newest array is counted as part of the orbit-representative array and discarded. However, if the graphs are not isoorphic, the newest array is stored and classified as part of the orbit representatives for the particular design UU. 4.3 The Backtrack Search Algorith Matheatically, we consider the idea of finding a FFd F ND with a design type U. This design specifies, aong other paraeters, the strength t of the array, which is the ain criterion looked in the search tree to drop a leaf. Definition 15. We define a partial iage to the set of distinct entries in a row by SS eeeeee = [F i1,, F ir ], 0 r. When r =, we call the sequence SS eeeeee as coplete. Fig. 2 Canonical graph of the orthogonal array F. 4.2 Algorith Description The construction process requires an existing orthogonal array with design type U ade using siple cobinatorial techniques. We then start adding the different sybols according to the specified design. By adding sybols, we create a new colun to the existing array until the new required OA is copleted. During the process of adding sybols, our algoriths check that the conditions of strength tt and sequence's lexicographical order are being et; otherwise, the entire OA is discarded. The aforeentioned process is carried out using the backtrack search algorith [11]. In addition to the previous criteria, we use our perutation group GG with the operations gg ρρ, gg γγ and gg ρρ to prune the tree when we find isoorphic graphs downstrea of the search tree (this stage According to definition 15, the backtrack search goes through the partial iages given by the sequence SS eeeeee. Within this search, it will use any knowledge of the design to prune the search tree. Note that in constructing a new design we totally order the sequences SS eeeeee so that we induce a lexicographical order for the set of partial iages. This eans that only the first point of each k-orbit has to be considered when extending SS eeeeee. Figure 3 shows the algorith we use to traverse a tree created to search orbits and their representatives. The search will go through soe of the eleents of the perutation group while skipping soe leaves according to the previously discussed criteria. 4.4 Isoorphis Classes Enueration Results In Table 1, we show soe exaples of results obtained by Man [1] which were corroborated with our approach. In addition, we introduce new isoorphis classes enuerated using the technique presented in this paper. The first colun of the table represents the run size NN for the different designs. The second colun corresponds to the actual design type UU according to the ultiplicity notation for autoorphis group orders. The third and fourth coluns of the table are the nuber of the autoorphis groups and their corresponding size respectively. We have indicated with an asterisk the newest designs we have found. 70

6 Fig. 3 Backtrack Search Algorith. 71

7 Table 1 Non-isoorphic OAs of strength three N Type # Size , 512 1, * * , , , , * * * * Note: new arrays indicated with an asterisk *. 5. CONCLUSION We have presented a atheatical and coputational ethod for constructing and enuerating strength-t orthogonal arrays given a fixed nuber of experients. The technique shown provided a feasible generic fraework and has been validated through both, the coparison of designs already listed by several different techniques, and the discovery of soe new ixed OAs. 6. ACKNOWLEDGEMENTS My deepest gratitude to Dr Scott H. Murray for his continuous support throughout the developent of this project. Many thanks also to Dr Judith Ascione for her very useful assistance during the early stages of preparing this anuscript. 7. REFERENCES [1] Nguyen M, Coputer-algebraic ethods for the construction of designs of experients. Technische Universiteit. Eindhoven, Eindhoven University Press, [2] Blanchard BS and Fabrycky WWJ, Systes engineering and analysis, Prentice-Hall international series in industrial and systes engineering, [3] Ross PJ, Taguchi techniques for quality engineering, McGraw-Hill Education (India), Pvt Liited, [4] Tsai JT, Robust optial-paraeter design approach for tolerance design probles, Engineering Optiization, 42(12): , [5] Duit DS and Foote RM, Abstract Algebra, Wiley, [6] Hedayat AS, Sloane NJA, and Stufken J, Orthogonal arrays: theory and applications, Springer series in statistics, Springer-Verlag, [7] Montgoery DC, Design and Analysis of Experients, Wiley, [8] Lekivetz R and Tang B, Multi-level orthogonal arrays for estiating ain effects and specified interactions, Journal of Statistical Planning and Inference, [9] David G and David B, Graphs for orthogonal arrays and projective planes of even order, SIAM Journal on Discrete Matheatics, 26(3): , [10] Schoen ED and Nguyen M, Enueration and classification of orthogonal arrays. Journal of Algebraic Statistics, [11] Hulpke A, Notes on Coputational Group Theory, Departent of Matheatics, Colorado State University, International Journal of GEOMATE, Jan., 2017, Vol. 12, Issue 29, pp MS No. 1308j received on July 15, 2015 and reviewed under GEOMATE publication policies. Copyright 2016, Int. J. of GEOMATE. All rights reserved, including the aking of copies unless perission is obtained fro the copyright proprietors. Pertinent discussion including authors closure, if any, will be published in Dec if the discussion is received by June Corresponding Author: Julio Roero 72