Unformatted text preview: Design of Experiments
Basic Concepts 1 Unit Objectives Introduce the terminology associated with Design of Experiments (DoE) Develop the procedures needed to identify Full and Partial Factorial tests Define the concept of Resolution with respect to Orthogonal Arrays Present Design of Experiment issues such as Blocking, SplitPlots, Nesting, and Mixture Experiments
2 1 Basic Concepts Experimental studies involve performing several tests at different combinations of factors and levels This could be done by performing arbitrarily chosen combination and recording the results This approach would probably lead to a bias in the final results, plus it may not be an efficient use of resources
3 Full Factorial Experiments I If the objective of a test is to determine which factors have an impact on a specific outcome then a more systematic set of experiments have to be performed Ideally, all combinations have to be tested multiple times (replications) This is called a Full Factorial Test
4 2 Full Factorial Experiments II
A full factorial set of experiment for three factors (A, B and C) at two levels (1 and 2) involves testing the following combinations at least twice A1B1C1, A1B1C2, A1B2C1, A1B2C2, A2B1C1, A2B1C2, A2B2C1, A2B2C2 The relationship is Total = Replications x LevelsFactors Replications Levels
5 Orthogonal Arrays
Factor A Factor B Factor C Replications x1 x2 x3 x4 1 1 1 1 1 2 x1 x2 x3 x4 1 2 1 x1 x2 x3 x4 1 2 2 x1 x2 x3 x4 x1 x2 x3 x4 2 1 1 2 2 2 1 2 2 2 1 2 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4
6 4x23 = 32 experiments 3 Completely Randomized Blocks These factor and level combinations are called treatments The order in which these treatments then has to be selected randomly to guarantees completely randomized blocks (CRB’s) This minimizes any bias that might exist if a specific sequence of factors is done in order
7 Partial Factorial Experiments I If it is too expensive to perform a full factorial set of experiments, a fraction of the experiments can be performed This is also called a Fractional Factorial Experiment (FFE) designs The results will be more suspect since some interaction effects will be mixed with effects associated with factors This is called confounding
8 4 Partial Factorial Experiments II
A 1/2r fractional factorial set of experiments with k factors is known as a 2kr design This is constructed by first identifying the full kr set of experiments Then select r high level interactions that are deemed to have no effect Use these to generate the levels for the remaining r factors
9 PFE Example
Determine the set of experiments for a 252 design with factors A, B, C, D, and E. Run 1 2 3 4 5 6 7 8 A – + – – + + – + B – – + – + – + + C – – – + – + + + D=AB + – – + + – – + E=BC + + – – – – + +
10 5 Partial Factorial Experiments II
Factor A 1 1 2 2 Factor B 1 2 1 2 Factor C 1 2 2 1 Replications x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 This is a partial factorial set of experiments for three factors. The degree of confounding is expressed as a resolution associated with the experiments (the higher the resolution the better).
11 Confounding In FFE’s “extra” factors are placed where interactions are observed It is not possible to distinguish between the effects linked to the factors or the associated interactions This phenomenon exists for all FFE’s and is called Confounding “Extra” factors should therefore be placed where interactions are weak
12 6 Partial Factorial Experiments III
Full Factorial
Notation
23 24 25 25 26 26 26 27 27 27 27 28 28 28 Partial Factorial
Fraction
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 2 2 2 4 2 4 8 2 4 8 No of Runs
8 16 32 32 64 64 64 128 128 128 128 256 256 256 No of Runs
4 8 16 8 32 16 8 64 32 16 8 64 32 16 Resolution
III IV V III VI IV III VII IV IV III V V IV 13 16 4 8 1/ 1/ 1/ 16 Resolution I Resolution is found by examining which factors and interactions share the same set of treatments (i.e. factor and level combinations) Level III
– No main aliased with a main – Main aliased with a two interaction – Two interaction aliased a two interaction
14 7 Resolution II Level IV – No main aliased with a main – No main aliased with a two interaction – Two interaction aliased a two interaction Level V – No main aliased with a main – No main aliased with a two interaction – No two interaction aliased with a two interaction
15 Suboptimal Issues Blocking – Attenuating affect of nuisance variables SplitPlots – Simplifying test procedures Nesting – Overcoming lack of repeatability Mixture Experiments – Used with constrained factors
16 8 Blocking I If a set of experiments has an identifiable nuisance variable, the effect that this variable has on the results has to be minimized This is done be performing identical sets of experiments for each “level” of the nuisance variable This is called Blocking
17 Blocking II Examples of nuisance variables that are used to create blocks are time of day, shift, operator, or spatial location For example, if the nuisance variable were operator, then each operator would perform the same set of experiments Within each block, however, the experiments would be randomized
18 9 Blocking III In blocking it is assumed that the nuisance variable does not interact with the factors being examined Latin and GraecoLatin squares are examples of blocking If there are only two nuisance variables a Latin square can be used The Latin square distributes the influence of the nuisance variable
19 Item Comparison
Focus Front Right Front Left Rear Right Rear Left A A A A Focus Front Right Front Left Rear Right Rear Left A A A A Focus Front Right Front Left Rear Right Rear Left A B C D Focus B B B B Mustang B B B B Mustang A B C D Focus C C C C Taurus C C C C Taurus A B C D Focus D D D D Thunderbird D D D D Thunderbird A B C D
20 10 Latin Square
To ensure that the results have a minimal bias, a Latin Square should be used to determine the appropriate combination of operating conditions that each item should experience.
Focus Front Right Front Left Rear Right Rear Left A B C D Mustang B C D A Taurus C D A B Thunderbird D A B C 21 GraecoLatin Squares I If a third nuisance variable exists then this can be included in the Latin Square by using Greek letters The result is a GraecoLatin Square Even in a GraecoLatin Square all experimental factors are associated with all nuisance variable levels, and even the nuisance variable levels are associated with each other.
22 11 GraecoLatin Squares II
Focus Front Right Front Left Rear Right Rear Left A B C D Mustang B A D C Taurus C D A B Thunderbird D C B A Tire Pressure – 25 psi – 30 psi – 35 psi – 40 psi
23 Blocking and Multiple Factors If multiple factors are to be examined each nuisance level should contain all treatments If multiple nuisance variables exist, the number of tests can be reduced by grouping extreme nuisance variables For example, variables A1 and B1 would be grouped together as would nuisance variables A3 and B3
24 12 Multiple Nuisance Variables If there are many nuisance variables to be blocked, then group the extreme cases of the variables into single blocks to reduce the number of tests If there are two nuisance variables (A and B) at three levels (1, 2 and 3) and only one block can be used then block A1 with B1, and A3 with B3
25 Blocking Example
Operator A Run 2 Run 3 Run 4 Run 1 Operator B Run 3 Run 1 Run 4 Run 2 Operator C Run 2 Run 4 Run 1 Run 3 Pressure 200 kPa 200 kPa 400 kPa 400 kPa Temperature 300 oC 400 oC 300 oC 400 oC
26 Run 1 Run 2 Run 3 Run 4 13 SplitPlots I In some cases it is difficult or expensive to change a specific factor thus making it difficult to completely randomize the set of experiments To address this issue the levels of the factors that are difficult to change can be done as a whole one at a time with the other factors being selected randomly within the block
27 SplitPlots II Identical to blocking except that the blocked element is a factor which might interact with the other factors The associated analysis is very similar to standard CRB analysis except that the degrees of freedom changes In most cases a CRB analysis is used
28 28 14 SplitPlot Example
Run 1 Run 2 Run 3 Run 4 Run 5 Run 6 Run 7 Run 8 Temperature 300 oC 300 oC 300 oC 300 oC 500 oC 500 oC 500 oC 500 oC Pressure 200 kPa 400 kPa 400 kPa 200 kPa 400 kPa 200 kPa 400 kPa 200 kPa Volume 0.25 m3 0.25 m3 0.75 m3 0.75 m3 0.75 m3 0.25 m3 0.25 m3 0.75 m3
29 Randomized SubBlocks Sub Nesting I Crossed Experiments are where all the levels of one factor are tested against identical levels of another factor Sometimes similar, but not identical, levels of one factor can only be compared against other factors This occurs only in the cases where the factor is an attribute
30 15 Nesting II Often occurs when the attribute is a physical entity, such as a person or machine In these cases the different levels of one factor may only be tested with similar physical entities, but not the same (identical) entity The associated analysis is a little different than a CRB analysis
31 Nesting Example
Muzzle Velocity Test Gun 1 Gun 2 Bullet 1 Bullet 2 Bullet 3 Bullet 1 Bullet 2 Bullet 3 The factor Bullet is said to be nested within the factor gun, since a different bullet 1, 2 or 3 has to be used for the different tests.
32 16 Mixture Experiment Example
Pb (100%) 20% Copper 40% Lead 40% Tin
60% 20% 80% 40% % Copper
60% % Lead
40% 80% 20% Cu (100%) 20% 40% 60% 80% % Tin Sn (100%)
33 Mixture Experiment Designs Unconstrained Mixtures – Simplex Design: (1,0,0) (0,1,0) (0,0,1) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2) – Augmented Simplex: Simplex plus (1/3,1/3,1/3) (2/3,1/6,1/6) (1/6, 2/3, 1/6) (1/6, 1/6, 2/3) Constrained Mixtures – Corners, halfway along edges, and central points
34 17 Mixture Experiment Example
Pb (100%) Simplex Points
20% 80% Augmentation Points
60% 40% % Copper
60% % Lead
40% 80% 20% Cu (100%) 20% 40% 60% 80% % Tin Sn (100%)
35 Summary
A full factorial set of experiments is needed to obtain a full understanding of an experimental study The number of experiments can be reduced by using a partial factorial set of experiments Constrained experimental designs can be addressed using blocking, splitplots, nesting, and mixture designs
36 18 ...
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 Fall '10
 JohnL
 Latin Square, nuisance variables, Partial Factorial Experiments

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