FURTHER ORTHOGONAL ARRAYS
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1 FURTHER ORTHOGONAL ARRAYS The Taguchi approach to experimental design Create matrices from factors and factor levels -either an inner array or design matrix (controllable factors) -or an outer array or noise matrix (uncontrollable factors) Here number of experiments required significantly reduced How? Orthogonal arrays 8/06/2007 ENGN8101 Modelling and Optimization 1
2 Orthogonal arrays - Trivial many versus vital few Orthogonal arrays - Matrix of numbers each column each factor or interaction each row levels of factors and interactions Main property: every factor setting occurs same number of times for every test setting of all other factors Allows for lots of comparisons Any two columns form a complete 2-factor factorial design Critical concept the LINEAR GRAPH 8/06/2007 ENGN8101 Modelling and Optimization 2
3 First example: L 4 array a half-replicate of a 2 3 experiment 4 experiments: factors level 1 or 2 3 factors? look at the linear graph: 2 nodes (columns 1 and 2) + 1 linkage (between 1 and 2 i.e. 2 factors + 1 interaction 8/06/2007 ENGN8101 Modelling and Optimization 3
4 The L 4 array cannot estimate 3 base factors (not yet!) also the nodes are different designs - associated with the degree of difficulty with changing the level of a particular factor Acknowledges that not all factors are easy to change Easy means easy to use as it only changes a minimum number of times if one factor is harder to change put it in column 1, as this only changes once Same for any size array 8/06/2007 ENGN8101 Modelling and Optimization 4
5 L 8 (2 7 ) array here 7 factors at 2 levels or 7 entities at 2 levels IMPORTANT! there are no 3-way interactions or higher represented by this method total replicate = 128 tests here only 8! 8/06/2007 ENGN8101 Modelling and Optimization 5
6 Also 2 linear graphs (templates for candidate experiments) 4 main effects + 3 interactions so long as one of the graphs fits your experiment - use the array! If not choose another or modify the graph (later) 8/06/2007 ENGN8101 Modelling and Optimization 6
7 Another template the L 9 (3 4 ) array here 4 factors, each at 3 levels should be 81 tests - actually 9 2 base factors only others confounded with interactions 8/06/2007 ENGN8101 Modelling and Optimization 7
8 Many others: L 16 (2 15 ) 5 base factors way interactions 8/06/2007 ENGN8101 Modelling and Optimization 8
9 L 27 (3 13 ) very powerful array 8/06/2007 ENGN8101 Modelling and Optimization 9
10 Also can have arrays for factors of varying number of levels e.g. L 18 (2 1 x 3 7 ) i.e. a hybrid (see later) 8/06/2007 ENGN8101 Modelling and Optimization 10
11 CASE STUDY T6 A consumer magazine subscription service has four factors A, B, C and D, each to be analysed at two levels. Also of interest are the interactions of BxC, BxD and CxD. Show the experimental design for this case 8/06/2007 ENGN8101 Modelling and Optimization 11
12 7 factors/interactions 2 levels A,B,C,D and BC, BD, CD 2 7? note: check the linear graphs! if they match use the L 8 approach factor A stand-alone i.e. no interaction of interest factor B,C,D base factors + 2 x 2-way interactions 1=B 2=C 3=BC 4=D 5=BD 6=CD 7=A - fits! 8/06/2007 ENGN8101 Modelling and Optimization 12
13 i.e. can we modify these graphs (templates) to account for other experimental designs? yes! 8/06/2007 ENGN8101 Modelling and Optimization 13
14 CASE STUDY T7 The rapid transport authority in a large metropolitan area has identified five factors, A, B, C, D and E, each to be investigated at 2 levels. Interactions AC and AD are also of interest. Determine an appropriate experimental design. 8/06/2007 ENGN8101 Modelling and Optimization 14
15 Here A, B, C, D, E + AC and AD i.e. 7 factors/interactions (5+2) candidate array = L 8 (2 7 ) currently not an option it gives 4 factors + 3 interactions we need 5 factors + 2 interactions we can modify the graph by breaking a link and creating a node from it preliminary allocation: interaction 6? (AE) 8/06/2007 ENGN8101 Modelling and Optimization 15
16 Pull it out and turn it into a node! i.e. the experiment now fits completely i.e. BUT/ factor B and interaction AE are now confounded therefore must assume AE = insignificant Orthogonal arrays lots of similar assumptions 8/06/2007 ENGN8101 Modelling and Optimization 16
17 Orthogonal arrays - graphs can be used to see what designs are possible either direct or modified Assumes no higher order interactions and that not all base factors or 2-way interactions are necessary plus-side 128 tests per replicate 8 tests! 8/06/2007 ENGN8101 Modelling and Optimization 17
18 HYBRID ORTHOGONAL ARRAYS i.e. technique for when not all factors have the same no. of levels CASE STUDY T8 A commercial bank has identified 5 factors (A-E) that have an impact on its volume of loans. There are 4 levels of factor A and 2 levels for each of the other factors. Determine an appropriate experimental design First find no. of degrees of freedom for each factor (always 1 less than no. of levels) i.e. A = 3; B=C=D=E = 1 total = 7 Same as for L 8 array (7 columns) use L 8 as our hybrid design template each column a 2-level interaction 1 degree of freedom/column 8/06/2007 ENGN8101 Modelling and Optimization 18
19 7 columns: 3 for A and 1 each for the other factors BUT/ which factor in which column? Consult the linear graph must identify a line that can be removed easily e.g. remove 1,2 and 3 to give 8/06/2007 ENGN8101 Modelling and Optimization 19
20 a new column 1 made up of old columns 1,2,3 7 columns now 5 a new A column sequentially index them i.e. rows 1,2, A=1 rows 3,4, A=2 rows 5,6, A=3 rows 7,8 A=4 8/06/2007 ENGN8101 Modelling and Optimization 20
21 Estimation of effects We have the experimental design. now run it! (r times) How many replicates? Often decided using noise factors Why include noise factors? To identify design factor levels that are least sensitive to noise i.e. robust 8/06/2007 ENGN8101 Modelling and Optimization 21
22 e.g. 4 factors: A, B, C, D + 3 noise factors: E, F, G need a design array (L 9 ) and a noise array (L 4 ) i.e. standard procedure 9 experiments run 4 times 8/06/2007 ENGN8101 Modelling and Optimization 22
23 2 extra columns Mean response = Ῡ mean of each set of 4 replicates S/N ratio = Z as given previous Ῡ and Z used in analysis phase i.e. the parameter design phase Taguchi approach uses simple plots to make inferences (ANOVA also possible) Main effect of a factor factor A levels 1,2,3 level 1 experiments 1,2,3 level 2 experiments 4,5,6 level 3 experiments 7,8,9 8/06/2007 ENGN8101 Modelling and Optimization 23
24 mean response when A is at level 1: A 1 = y 1 + y y 3 etc Another example; factor B at level 3: A 1 = y 1 + y y 3 For each factor 3 points now plot!! 3 types of plot: 8/06/2007 ENGN8101 Modelling and Optimization 24
25 Type a: Type b: effect not significant i.e. not worth bothering with (?) effect = non-linear best selection region where curve is flattest (i.e. minimum gradient i.e. minimum variability with response variable here level 2 is the most robust setting Type c: effect = linear here factor = adjustment parameter gradient is constant constant variation but can change mean response easily 8/06/2007 ENGN8101 Modelling and Optimization 25
26 Can repeat the procedure with interaction effects: Interaction of BxC and CASE STUDY T9 ( BxC) ( BxC) 1 2 = = y + y + y + y y + y + y + y Various components of a drug for lung cancer have positive and negative effects depending on the amount used. Scientists have identified four independent factors that seem to affect the performance of the drug. 8/06/2007 ENGN8101 Modelling and Optimization 26
27 4 factors x 3 levels L 9 (3 4 ) need to modify the array assume no interaction factors Now run tests (target value = 0) Only 1 replicate no noise factors possible 8/06/2007 ENGN8101 Modelling and Optimization 27
28 Main effects: etc A1 = = A2 = = A3 = = now plot 3 8/06/2007 ENGN8101 Modelling and Optimization 28
29 So what? B and D non-linear A almost linear C linear For a robust system set B and D to level 2 to reduce variability Then move the response value to zero using adjustment factors i.e. set A and C to level 2 optimal setting = A=B=C=D=2 NOTE/ not one of our original experiments! This is the essence of Taguchi parameter design - to find the best parameter settings using 2-stage 2 optimization and indirect experimentation of course further testing will confirm this 8/06/2007 ENGN8101 Modelling and Optimization 29
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