An Algorithm For Constructing Orthogonal And Nearly .
An Algorithm for Constructing Orthogonaland Nearly-Orthogonal Arrays WithMixed Levels and Small RunsHongquan XuDepartment of StatisticsUniversity of CaliforniaLos Angeles, CA 90095( hqxu@stat.ucla.edu)Orthogonal arrays are used widely in manufacturing and high-technology industries for quality andproductivity improvement experiments. For reasons of run size economy or exibility, nearly-orthogonalarrays are also used. The construction of orthogonal or nearly-orthogonal arrays can be quite challenging. Most existing methods are complex and produce limited types of arrays. This article describesa simple and effective algorithm for constructing mixed-level orthogonal and nearly-orthogonal arraysthat can construct a variety of small-run designs with good statistical properties ef ciently.KEY WORDS:1.D-optimality; Exchange algorithm; Interchange algorithm; J2 -optimality.INTRODUCTIONConsider an experiment to screen factors that may in uencethe blood glucose readings of a clinical laboratory testingdevice. One two-level factor and eight three-level factorsare included in the experiments. The nine factors (Wuand Hamada 2000, table 7.3) are (A) wash (no or yes),(B) microvial volume (2.0, 2.5, or 3.0 mL), (C) caras H2 Olevel (20, 28, or 35 mL), (D) centrifuge speed (2,100, 2,300,or 2,500 rpm), (E) centrifuge time (1.75, 3, or 4.5 minutes),(F) sensitivity (.10, .25, or .50), (G) temperature (25, 30,or 37 C), (H) dilution ratio (1:51, 1:101, or 1:151), and (I)absorption (2.5, 2, or 1.5). To ensure that all the main effectsare estimated clearly from one another, it is desirable touse an orthogonal array (OA). The smallest OA found forone two-level factor and eight three-level factors requires36 runs. However, the scientist wants to reduce the cost ofthis experiment and plans to use an 18-run design. A goodsolution then is to use an 18-run nearly-orthogonal array(NOA).The concept of OA dates back to Rao (1947). OAs havebeen used widely in manufacturing and high-technologyindustries for quality and productivity improvement experiments, as evidenced by many industrial case studies and recentdesign textbooks (Myers and Montgomery 1995; Wu andHamada 2000). Applications of NOAs have been describedby Wang and Wu (1992), Nguyen (1996b), and the referencescited therein.Formally, an OA of strength two, denoted by OA4N 1 s1 sn 5, is an N n matrix of which the ith column has si levelsand for any two columns all of their level combinations appearequally often. An OA is mixed if not all si ’s are equal. AnNOA, denoted by OA0 4N 1 s1 sn 5, is optimal under the J2 criterion (de ned in Sec. 2.1). From an estimation standpoint, allof the main effects of an OA are estimable and orthogonal toeach other, whereas all of the main effects of an NOA are stillestimable, but some are partially aliased with others. Becausebalance is a desired and important property in practice, inthis article only balanced OA0 4N 1 s1 sn 5 are considered, inwhich all levels appear equally often for any column. When anarray is used as a factorial design, each column is assigned toa factor and each row corresponds to a run. Here the termsof “array” and “design,” “row” and “run,” and “column” and“factor” are freely exchanged.The purpose of this article is to present a simple andeffective algorithm for constructing OAs and NOAs withmixed levels and small runs. The algorithm can ef cientlyconstruct various designs with good statistical properties.Section 2 introduces the concept of J2 -optimality and otheroptimality criteria. Section 3 describes an algorithm forconstructing mixed-level OAs and NOAs. Section 4 gives theperformance and compares the algorithm with others in termsof speed and ef ciency. Section 5 revisits the blood glucoseexperiment, and Section 6 gives concluding remarks.2.OPTIMALITY CRITERIAA combinatorial criterion, J2 -optimality, is introduced inSection 2.1. This criterion has the advantages of conveniencefor programming and ef ciency for computation. The statistical justi cation of J2 -optimality and other optimality criteriais given in Section 2.2.2.1The Concept of J2 -OptimalityFor an N n matrix d D 6xik 7, weight wk 0 is assignedfor column k, which has sk levels. For 1 µ i1 j µ N , let„i1 j 4d5 DnXwk „4xik 1 xjk 51(1)kD1where „4x1 y5 D 1 if x D y and 0 otherwise. The „i1 j 4d5 valuemeasures the similarity between the ith and jth rows of d.356 2002 American Statistical Association andthe American Society for QualityTECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4DOI 10.1198/004017002188618554
ORTHOGONAL AND NEARLY-ORTHOGONAL ARRAYSIn particular, if wk D 1 is chosen for all k, then „i1 j 4d5 is thenumber of coincidences between the ith and jth rows. De ne357The coincidence matrix 4„i1 j 4d55 of the 12 rows A design is J2 -optimal if it minimizes J2 . Obviously, by minimizing J2 4d5, it is desired that the rows of d be as dissimilaras possible. The following lemma shows an important lowerbound of J2 emma 1. For an N n matrix d whose kth column hassk levels and weight wk 2 4d5 DX6„i1 j 4d572 01µi jµNJ2 4d5 ¶ L4n5"³ 2 ³ n nXXƒ1ƒ1ƒ12D2Nsk wk C4sk ƒ 154Nsk wk 5kD1ƒN³nXkD1 2 #wk1kD1(2)and the equality holds if and only if d is an OA.The proof is given in Appendix A. From Lemma 1, an OAis J2 -optimal with any choice of weights if it exists, whereasan NOA under J2 -optimality may depend on the choice ofweights.Example 1. Consider the 12 10 matrix given in Table 1.The rst column has three levels, and the other nine columnshave two levels each. For illustration, wk D 1 is chosen for allk. First, consider a design comprising the rst ve columns.and J2 is the sum of squares of the elements above the diagonal. It is easy to verify that J2 D 330 and that the lowerbound in (2) is also 330 for one three-level and four two-levelcolumns with wk D 1. Therefore, the rst ve columns forman OA4121 31 24 5, because the J2 value equals the lower bound.Next, consider the whole array, comprising all 10 columns.Simple calculation shows that J2 D 11284 and that the lowerbound in (2) is 1,260. Therefore, the whole array is not anOA, because the J2 value is greater than the lower bound.Now consider the change in the J2 value if a column isadded to d or if two symbols are switched in a column. Ifa column c D 4c1 1 : : : 1 cN 50 is added to d and dC is the newN 4n C 15 design, and if c has sk levels and weight wk , thenfor 1 µ i1 j µ N ,„i1 j 4dC 5 D „i1 j 4d5 C wk „i1 j 011NOTE: The ’ rst ’ ve columns form an OA(121 3 1 2 4 ). The pairs of columns (1, 6) and (1, 10)are nonorthogonal and have an A 2 value of .167; the pairs of columns (2, 9), (3, 7), (4, 8),and (6, 10) are nonorthogonal and have an A 2 value of .111; and all other pairs of columnsare orthogonal.(3)where „i1 j 4c5 D „4ci 1 cj 5. In addition, if the added column cis balanced, then it is easy to show thatJ2 4dC 5 D J2 4d5 C 2wkTable 1. OA0 ( 12, 31 29 )1X„i1 j 4d5„i1 j 4c51µi jµNC 2ƒ1 Nwk2 4Nskƒ1 ƒ 150(4)The summation in the second term on the right side of (4) doesnot involve any multiplication, because „i1 j 4c5 is either 0 or 1.Therefore, calculating J2 4dC 5 as in (4) is much faster than bytaking the sum of squares of all „i1 j 4dC 5 in (3). Now supposethat two distinct symbols, ca 6D cb , in rows a and b in the addedcolumn are switched. Then all „i1 j 4c5 are unchanged, exceptthat „a1 j 4c5 D „j1 a 4c5 and „b1 j 4c5 D „j1 b 4c5 are switched forj 6D a1 b. Hence J2 4dC 5 is reduced by 2wk ã4a1 b5, whereXã4a1b5 D6„a1j 4d5 ƒ „b1j 4d576„a1j 4c5 ƒ „b1j 4c570 (5)1µj 6Da1bµNCalculation of ã4a1 b5 involves no multiplication, becauseboth „a1 j 4c5 and „b1 j 4c5 are either 0 or 1. These formulasprovide an ef cient way to update the J2 value and are usedin the algorithm.TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4
3582.2HONGQUAN XUOther Optimality Criteria andStatistical Justi’ cation of J2 -OptimalityTo gain an understanding of the statistical justi cation ofJ2 -optimality, recall other optimality criteria. It is well knownthat an s-level factor has s ƒ 1 degrees of freedom. Commonlyused contrasts are from orthogonal polynomials, especially forquantitative factors. For example, the orthogonal polynomialscorresponding to levels 0 and 1 of a two-level factor are ƒ1and C1, and the orthogonal polynomials corresponding to levels 0, 1, and 2 of a three-level factor are ƒ1, 0, and C1 forlinear effects and 1, ƒ2, and 1 for quadratic effects.For an N n matrix d D 6xik 7, whose kth column has sklevels, consider the main-effects modelY D ‚0 1 C X1 ‚1 C 1where Y is the vector of N observations, ‚0 is the generalmean, ‚1 is the vector of all the main effects, 1 is the vector of1s, X1 is the matrix of contrast coef cients for ‚1 , and is the1 : : : 1 xm 5vector of independent random errors. Let X1 D 4x1Pand X D 4x1 x1 1 : : : 1 xm xm 5, where m D 4si ƒ 15.In the literature, d is known as the design matrix and X1is the model matrix (of the main-effects model). A designis D-optimal if it maximizes —X 0 X —. It is well known that—X 0 X — µ 1 for any design and that —X 0 X — D 1 if and only if theoriginal design d is an OA. Wang and Wu (1992) proposedthe D criterionD D —X 0 X —1 m(6)to measure the overall ef ciency of an NOA. Note thatR D X 0 X is the correlation matrix of m columns of X 1 .A good surrogate for the D criterion is the 4M 1 S5 criterion(Eccleston and Hedayat 1974). A design is 4M 1 S5-optimalif it maximizes tr4X 0 X5 and minimizes tr64X 0 X5 2 7 amongthose designs that maximize tr4X 0 X5. The 4M 1 S5 criterion ischeaper to compute than the D criterion and has been usedin the construction of computer-aided designs (see, e.g., Lin1993; Nguyen 2001). Because all diagonal elements of X 0 Xare 1s, the 4M 1 S5 criterion reduces to the minimization oftr64X 0 X52 7, which is the sum of squares of elements of X 0 X,or, equivalently, to the minimization of the sum of squares ofoff-diagonal elements of X 0 X. This minimization leads to thefollowing concept of A2 -optimality.Formally, if X 0 X D 6rij 7, letX 2A2 Drij 1i jwhich measures the overall aliasing (or nonorthogonality)between all possible pairs of columns. In particular, for atwo-level design, A2 equals the sum of squares of correlationbetween all possible pairs of columns, and therefore it isequivalent to the popular ave4s 2 5 criterion in the contextof two-level supersaturated designs. A design is A2 -optimalif it minimizes A2 . This is a good optimality criterion forNOAs because A2 D 0 if and only if d is an OA. Further,A2 -optimality is a special case of the generalized minimumaberration criterion proposed by Xu and Wu (2001) forassessing nonregular designs.The statistical justi cation for J2 -optimality arises from thefollowing lemma, which shows an important identity relatingthe J2 and A2 criteria.TECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4Lemma 2. For a balanced design d of N runs and n factors, if the weight equals the number of levels for each factor4i.e., wk D sk 5, thenhXX 2 iJ2 4d5 D N 2 A2 4d5 C 2ƒ1 N Nn4n ƒ 15 C Nsk ƒsk 0The proof is given in Appendix A. For convenience, thechoice of wk D sk is called natural weights. J2 -optimality withnatural weights is equivalent to A2 -optimality and thus is agood surrogate for D-optimality.Advantages of the use of J2 over D, 4M 1 S5, A2 and ave4s 2 5as an objective function include the following:1. It is simple to program. J2 works with the design matrix,whereas all other criteria work with the model matrix.2. It is cheap to compute. Neither the calculation of „i1 j 4d5in (1) nor that of „i1 j 4dC 5 in (3) involves any multiplication,because both „4xik 1 xjk 5 and „i1 j 4c5 are either 0 or 1, and thisspeeds up the algorithm.3. It works with columns of more than two levels. Notethat the NOA algorithm of Nguyen (1996b) works only withtwo-level columns. To construct an OA0 4181 21 38 5, for example, Nguyen has to use a separate blocking algorithm (seeNguyen 2001) to divide an OA4181 21 37 5 into three blocks.4. It works with any choice of weights. By choosingproper weights, one can construct different types of NOAswith a single algorithm. Note that to construct two types ofOA0 4121 31 29 5’s, Nguyen (1996b) has to code the three-levelcolumn differently in his NOA algorithm. This advantage isdiscussed in more detail at the end of Section 4.2.5. It is very ef cient when the number of runs is less thanthe number of parameters, as in the case of supersaturateddesigns.3.AN ALGORITHMThe basic idea of the algorithm is to add columns sequentially to an existing design. The sequential operation isadopted for speed and simplicity. This operation avoidsan exhaustive search of columns for improvement, whichcould be complex and inef cient in computation. The twooperations when adding a column are interchange andexchange. The interchange procedure, also called the pairwiseswitch, switches a pair of distinct symbols in a column. Foreach candidate column, the algorithm searches all possibleinterchanges and makes an interchange that reduces J2 themost. The interchange procedure is repeated until a lowerbound is achieved or until no further improvement is possible.The exchange procedure replaces the candidate column by arandomly selected column. This procedure is allowed to repeatat most T times if no lower bound is achieved. The valueof T depends on the orthogonality of the previous design.If the previous design is an OA, then T D T1 ; otherwise,T D T2 , where T1 and T2 are two constants controlled by theuser. With any speci ed weights w1 1 : : : 1 wn , the algorithmconstructs an OA0 4N 1 s1 sn 5, in which the rst n0 columnsform an OA4N 1 s1 : : : sn0 5.The algorithm proceeds as follows:1. For k D 11 : : : 1 n, compute the lower bound L4k5 according to (2).
ORTHOGONAL AND NEARLY-ORTHOGONAL ARRAYS2. Specify an initial design d with two columns, 401 : : : 1 0111 : : : 1 11 : : : 1 s1 ƒ 11 : : : 1 s1 ƒ 15 and 401 : : : 1 s2 ƒ 11 01 : : : ,s2 ƒ 11 : : : 1 01 : : : 1 s2 ƒ 15. Compute „i1 j 4d5 and J2 4d5 by de nition. If J2 4d5 D L425, then let n0 D 2 and T D T1 ; otherwise,let n0 D 0 and T D T2 .3. For k D 31 : : : 1 n, do the following:a. Randomly generate a balanced sk -level column c.Compute J2 4dC 5 by (4). If J2 4dC 5 D L4k5, go to (d).b. For all pairs of rows a and b with distinct symbols, compute ã4a1 b5 as in (5). Choose a pair of rowswith largest ã4a1 b5 and exchange the symbols in rowsa and b of column c. Reduce J2 4dC 5 by 2wk ã4a1 b5. IfJ2 4dC 5 D L4k5, then go to (d); otherwise, repeat (b) untilno further improvement is made.c. Repeat (a) and (b) T times and choose a column cthat produces the smallest J2 4dC 5.d. Add column c as the kth column of d, let J2 4d5 DJ2 4dC 5, and update „i1 j 4d5 by (3). If J2 4d5 D L4k5, then letn0 D k; otherwise, let T D T2 .4. Return the nal N n design d, of which the rst n0columns form an OA.This is an example of a columnwise algorithm. As notedby Li and Wu (1997), the advantage of columnwise instead ofrowwise operation is that the balance property of a design isretained at each iteration. A simple way of adding an s-levelcolumn is to choose a best column from all possible candidate columns. However, it is computationally impossible toenumerate all possiblecandidate columns if the run size N 3is not small: NN 2 balanced columns for s D 2 and NN 3 2NN 3balanced columns for s D 3. The numbers growexponentially 24D 217041 156; 32D 6011 0801 390;with16 N ; for example, 12 18 1227 18D 171 1531 136; and 18 9203 1011 . The inter126change and exchange procedures used in the algorithm yielda feasible approach to minimizing J2 for computational ef ciency. An interchange operation searches N 2 4 columns fors D 2, N 2 3 columns for s D 3, and fewer than N 2 2 columnsfor any s. The interchange procedure usually involves a few(typically less than six) iterations. Compared with the sizeof all candidate columns, the interchange operation searchesonly a rather small portion of the whole space. Thus it is anef cient local learning procedure, but often ends up with alocal minimum. For this reason, global exchange proceduresare also incorporated into the algorithm to allow the search tomove around the whole space and not be limited to a smallneighborhood. As discussed later, the global exchange procedure with moderate T1 and T2 improves the performance ofthe algorithm tremendously.The values of T1 and T2 determines the speed and performance of the algorithm. A large Ti value allows the algorithmto spend more effort in searching for a good column, whichtakes more time. The choice of T1 and T2 depends on the typeof design to be constructed. For constructing OAs, a moderateT1 , say 100, is recommended and T2 can be 0; for constructingNOAs, moderate T1 and T2 are recommended. More detailsare given in the next section.Remark 1. Both interchange and exchange algorithmshave been proposed by a number of authors for various purposes. (See Nguyen 1996a and Li and Wu 1997 in the contextof constructing supersaturated designs.)359Remark 2. The performance of the algorithm may dependon the order of levels. Experience suggests that it is mosteffective if all levels are arranged in a decreasing order (i.e.,s1 ¶ s2 ¶ ¶ sn ), because the number of balanced columnsis much larger for a higher level than a lower level.Remark 3. The speed of the algorithm is maximizedbecause only integer operations are required if integralweights are used. For ef ciency and exibility, the algorithmis implemented as a function in C and can be called from S.Both C and S source codes are available from the author onrequest.4.PERFORMANCE AND COMPARISONThis section reports the performance and comparison of thealgorithm with others for the construction of OAs and NOAs.4.1Orthogonal ArraysIn the construction of OAs, the weights can be xed atwi D 1, and T2 should be 0 because it is unnecessary to continue adding columns if the current design is not orthogonal.Here the choice of T1 is studied in more detail, because itdetermines the speed and performance of the algorithm.The algorithm is tested with four choices of T1 : 11 101 100and 11000. For each OA and T1 , the algorithm is repeated11000 times with different random seeds on a Sun SPARC400-MHz workstation. It either succeeds or fails in constructing an OA each time. Table 2 shows the success rate andthe average time in seconds over 1,000 repetitions. In theconstruction of a mixed-level OA, as stated in Remark 2,the levels are arranged in a decreasing order. Table 2 showsclearly the trade-off between the success rate and speed, whichdepends on the choice of T1 . The success rate increases and thespeed decreases as T1 increases. A good measure is the number of OAs constructed per CPU time. The algorithm is leastef cient for T1 D 1 and is more ef cient for T1 D 10 or 100than T1 D 11000. Overall, the choice of T1 D 100 balancessuccess rate and speed and so is generally recommended.The construction of OAs continues to be an active researchtopic since Rao (1947) introduced the concept. Constructionmethods include combinatorial, geometrical, algebraic, coding theoretic, and algorithmic approaches. State-of-the-art construction of OAs has been described by Hedayat, Sloane, andStufken (1999). Here the focus is on algorithmic constructionand comparison with existing algorithms.Many exchange algorithms have been proposed for constructing exact D-optimal designs. (For reviews, see Cookand Nachtsheim 1980 and Nguyen and Miller 1992.) Thesealgorithms can be used to construct OAs; however, they areinef cient, and the largest OA constructed and published sofar is OA4121 211 5 (Galil and Kiefer 1980). By modifying theexchange procedure, Nguyen (1996a) proposed an interchangealgorithm for constructing supersaturated designs. His program can be used to construct two-level OAs; the largest OAconstructed and published is OA4201 219 5.Global optimization algorithms, including simulated annealing (Kirkpatrick, Gelatt, and Vecchi 1983), thresholdingaccepting (Dueck and Scheuer 1990), and genetic algorithmsTECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4
360HONGQUAN XUTable 2. Performance in the Construction of OAsT1 D 1Array4OA(91 3 )OA(121 211 )OA(161 81 28 )OA(161 215 )OA(161 45 )OA(181 37 21 )OA(181 61 36 )OA(201 219 )OA(201 51 28 )OA(241 223 )OA(241 41 220 )OA(241 31 216 )OA(241 121 212 )OA(241 41 31 213 )OA(241 61 41 211 )OA(251 56 )OA(271 91 39 )OA(271 313 )OA(281 227 )OA(321 161 216 )OA(321 81 42 218 )OA(401 201 220 )NOTE:T1 D 10T1 D 100T1 D 05441015850644130972The entries in the columns are the success rate of constructing an OA and the average time in seconds per repetition.(Goldberg 1989), may be used to construct OAs. These algorithms often involve a large number of iterations and are veryslow to converge. These algorithms have been applied to manyhard problems and are documented to be powerful. However,they are not effective in the construction of OAs (Hedayatet al. 1999, p. 337); for example, thresholding acceptingalgorithms of Fang, Lin, Winker, and Zhang (2000) and Ma,Fang, and Liski (2000) failed to produce any OA4271 313 5 orOA4281 227 5.DeCock and Stufken (2000) proposed an algorithm for constructing mixed-level OAs via searching some existing twolevel OAs. Their purpose is to construct mixed-level OAs withas many two-level columns as possible, and their algorithmsucceeded in constructing several new large mixed-level OAs.In contrast, the purpose in the present article is to constructas many nonisomorphic mixed-level OAs (with small runs)as possible, for which the proposed algorithm is more exible and effective. For example, the proposed algorithm isquite effective in constructing an OA4201 51 28 5 that is knownto have maximal two-level columns whereas the algorithmof DeCock and Stufken fails to produce any OA4201 51 27 5.Furthermore, the proposed algorithm successfully constructsseveral new 36-run OAs not constructed by their algorithm.Appendix B lists nine new 36-run OAs.It is interesting to have some head-to-head timing comparisons between this and other algorithms. A Fedorov exchangealgorithm and Nguyen’s NOA algorithm are chosen forcomparison. Cook and Nachtsheim (1980) reported that theFedorov exchange algorithm produces the best result but takesthe longest CPU time among several D-optimal exchangealgorithms. The Fortran source code due to Miller and Nguyen(1994) is used for the Fedorov algorithm, downloaded fromStatLib (http://www.lib.stat.cmu.edu). The Nguyen algorithmis implemented by replacing ave4s 2 5 with J2 for conveniencebecause no source code is available. This modi cation willTECHNOMETRICS, NOVEMBER 2002, VOL. 44, NO. 4affect the speed, but not the ef ciency in constructing OAs.Table 3 shows the comparisons of the algorithms in terms ofspeed and ef ciency. All algorithms were compiled and runon an iMac PowerPC G4 computer for a fair comparison.The iMac computer has a 867-MHz CPU, about three timesfaster than the Sun workstation described earlier. In thesimulation, the Fedorov algorithm was repeated 1,000 timesfor OA4121 211 5 and OA4161 215 5 because it is very slow,and other algorithms were repeated 10,000 times for allarrays. Table 3 clearly shows that the proposed algorithmperforms the best and the Fedorov algorithm performs theworst in terms of both speed and ef ciency. The Fedorovalgorithm is slow because it uses an exhaustive search ofpoints for improvement and uses D-optimality as the objectivefunction, which involves real-valued matrix operations. The(modi ed) Nguyen algorithm is slower than the proposedalgorithm, because the former uses a nonsequential approachand the latter uses a sequential approach. The nonsequentialapproach stops only when no swap is made on any columnfor consecutive n times (where n is the number of columns),whereas the sequential approach stops when it fails to addan orthogonal column for consecutive T1 times. When T1is less than n, the sequential approach stops earlier in thecase of failure. This explains why the sequential approachis faster. Furthermore, the high success rate of the proposedalgorithm shows that the sequential approach is more ef cientthan the nonsequential approach. Note that with the increasedcomputer power, the Fedorov algorithm succeeds in generatingsome OA4161 215 5’s, whereas the Nguyen algorithm still failsto generate any OA4241 223 5 in 10,000 repetitions.4.2Nearly-Orthogonal ArraysWang and Wu (1992) systematically studied NOAs andproposed some general combinatorial construction methods.Nguyen (1996b) proposed an algorithm for constructing
ORTHOGONAL AND NEARLY-ORTHOGONAL ARRAYS361Table 3. Comparison of Algorithms in Terms of Speed and Ef’ ciencyFedorovArrayOA(121 211 )OA(161 215 )OA(201 219 )OA(241 223 )NguyenAuthor (T1 D 1)Author (T1 D e.0005.0026.0096.0189NOTE: The entries in the columns are the success rate of constructing an OA and the average time in seconds per repetition. TheFedorov algorithm is due to Miller and Nguyen (1994), and the Nguyen algorithm is implemented with J2 -optimality.NOAs by adding two-level columns to existing OAs. Ma et al.(2000) proposed two algorithms for constructing NOAs byminimizing some combinatorial criteria via the thresholdingaccepting technique. Here the proposed algorithm is used toconstruct J2 -optimal mixed-level NOAs and compare themwith others. Table 4 shows the comparison of NOAs in termsof A2 and D optimality. The arrays are chosen according toA2 -optimality, that is, J2 -optimality with natural weights. Ofthe designs with the same A2 value, the one with the highestD ef ciency is chosen. Orthogonal polynomial contrasts areused to calculate the D ef ciency as in (6). The number ofnonorthogonal pairs, Np, is also reported for reference. Inn nthe construction of an OA0 4N 1 s1 1 s2 2 5, all s1 -level columnswith weight s1 are entered ahead of s2 -level columns withweight s2 . The algorithm is very ef cient; most arrays can beobtained within seconds with the choice of T1 D T2 D 100.The advantage of the proposed algorithm is clear fromTable 4. Among all cases, the arrays from this algorithm havethe smallest A2 value and the largest D ef ciency. Amongthe algorithms, the thresholding accepting algorithms of Maet al. are most complicated but perform the worst. The poorperformance of these algorithms again suggests that globaloptimization algorithms are not effective in the constructionTable 4.Comparison of NOAs With Run Size µ24Wang and WuArrayA2OA0 (61 31 23 )OA0 (101 51 25 )OA0 (121 41 34 )OA0 (121 23 34 )OA0 (121 61 25 )OA0 (121 61 26 )OA0 (121 31 29 )OA0 (121 21 35 )OA0 (121 27 32 )OA0 (121 25 33 )OA0 (151 51 35 )OA0 (181 21 38 )OA0 (181 37 23 )OA0 (181 91 28 )OA0 (201 51 215 )OA0 (241 81 38 )OA0 (241 31 221 )OA0 (241 61 215 )OA0 (241 61 218 )OA0 (241 21 311 )OA0 (241 31 47 5044.871.5945521of OAs and NOAs. The Nguyen arrays are competitivein terms of D and Np; however, Nguyen’s algorithm canconstruct only a small number of arrays, because it worksonly with two-level columns. Among the designs listed inTable 4 is a new OA0 4201 51 215 5 that is better than Nguyen’sin terms of both A2 and D. That array has 19 nonorthogonalpairs of columns, and his has 25 nonorthogonal pairs; onthe other hand, that array has 7 orthogonal columns (the rst 7) and his has 8 orthogonal columns (the rst 8).Moreover, two arrays, OA0 4121 31 29 5 and OA0 4241 31 221 5,are better than Nguyen’s in terms of A2 . In terms of Np,the arrays OA0 4121 31 29 5 and OA0 4201 51 215 5 are better andOA0 4181 21 38 5 and OA0 4241 31 221 5 are worse than his. Forreference, these arrays are listed in Tables 1, 5, 6, and 7.The proposed algorithm has an additional feature: Weightscan be used to control the structure of NOAs. For example,consider the construction of an OA0 4121 31 29 5. If the practitioner is more concerned with a three-level factor, then it isdesirable to have an NOA in which the three-level column isorthogonal to all two-level columns. If the practitioner is moreconcerned with the two-level factors, it is desirable to have anNOA in which all two-level columns are orthogonal to eachother. Wang and Wu (1992) referred to such designs as typeMa et 7.933.877.909.877.882.967.970.985.925.897.968.
Orthogonal arrays are used widely in manufacturing and high-technology industries for quality and productivity improvement experiments. For reasons of run size economy or ‘ exibility, nearly-orthogonal arrays are also u
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
Creating Orthogonal Array. As we will explain shortly, the technique proposed in this paper usually requires dif-ferent orthogonal arrays for di erent problems. e con-struction of orthogonal array is not a trivial task since we do not know whether an orthogonal array of given size exists. Many
6-Pair Orthogonal right angle header connector solution. The connectors enable efficient implementation of Direct-Mate orthogonal and midplane orthogonal architectures. Orthogonal architecture solutions eliminate long, complex traces, via stub effects, simplify signal links and reduce backplane layer count.
10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan
service i Norge och Finland drivs inom ramen för ett enskilt företag (NRK. 1 och Yleisradio), fin ns det i Sverige tre: Ett för tv (Sveriges Television , SVT ), ett för radio (Sveriges Radio , SR ) och ett för utbildnings program (Sveriges Utbildningsradio, UR, vilket till följd av sin begränsade storlek inte återfinns bland de 25 största
Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
LÄS NOGGRANT FÖLJANDE VILLKOR FÖR APPLE DEVELOPER PROGRAM LICENCE . Apple Developer Program License Agreement Syfte Du vill använda Apple-mjukvara (enligt definitionen nedan) för att utveckla en eller flera Applikationer (enligt definitionen nedan) för Apple-märkta produkter. . Applikationer som utvecklas för iOS-produkter, Apple .
uplifting tank and the plastic deformation of the bottom plate at the shell-to-bottom juncture in the event of earthquake, the design spectrum for sloshing in tanks, the design pressure for silos, and the design methods for the under-ground storage tanks as well. The body of the recommendation was completely translated into English but the translation of the commentary was limited to the .
