Terms and Definitions
Quality Characteristic – Quality Characteristic (QC) generally refers to the measured results of the experiment. The QC can be a single criterion such as pressure, temperature, efficiency, hardness, surface finish, etc., or a combination of several criteria together into a single index. QC also refers to the nature of the performance objectives such as “bigger is better”, “smaller is better” or “nominal is the best”.
In most industrial applications, QC consists of multiple criteria. For example, an experiment to study a casting process might involve evaluating cast specimens in terms of (a) hardness, (b) visual inspection of the surface, and (c) number of cavities. To analyze results, readings of evaluation under each of these three criteria for each test sample can be used to determine the optimum. The optimum conditions determined by using the results of each criterion may or may not yield the same factor combination for the optimum. Therefore, a weighted combination of the results under different criteria into a single quantity may be highly desirable. While combining the results of different criteria, they must first be normalized and then made to be of type ‘smaller is better’ or ‘bigger is better’.
When quality characteristic (QC) consists of, say, three criteria, an overall evaluation criteria (OEC) can be constructed as:
OEC = (X1/X1ref.)W1 + (X2/X2ref.)W2 + (X3/X3ref.)W3
where X = evaluation under a criterion
Xref = a reference (maximum) value of reading
W = weighting factor of the criterion (in %)
The use of OEC as the result of an experimental sample instead of several readings from all criteria offers an objective method of determining the optimum condition based on overall performance objectives.
When there are multiple criteria of evaluation, the experimenter can analyze the experiments based on readings under one category at a time as well as by using the OEC. If the individual outcomes differ from each other, the optimum obtained by using OEC as a result should be preferred.
What are the factors and levels?
Factors are
– design parameters that influence the performance.
– input that can be controlled.
– included in the study to determine their influence and control upon the most desirable performance.
Example: In a cake baking process the factors are; Sugar, Flour, Butter, Egg, etc.
Levels are
– Values that a factor assumes when used in the experiment
Example: As in the above cake-baking process the levels of sugar and flour could be:
2 pounds, 5 pounds, etc. (Continuous level) type 1, type 2, etc. (Discrete level)
LIMITS: Number of factors: 2 -63, number of levels: 2, 3, and 4.
INTERACTION BETWEEN FACTORS
Two factors (A and B) are considered to have interaction between them when one influences the effect of the other factor respectively.
Consider the factors “temperature” and “humidity” and their influence on comfort level. If the temperature is increased by, say 20 degrees F, the comfort level decreases by, say 30% when humidity is kept at 90%. On a different day, if the temperature is raised the same amount at a humidity level of 70%, the comfort level is reported to drop only by 10%. In this case, the factors “temperature” and “humidity” are interacting with each other.
Interaction:
– is an effect (output) and does not alter the trial condition.
– can be determined even if no column is reserved for it.
– can be fully analyzed by keeping appropriate columns empty.
– affects the optimum condition and the expected result.
NOISE FACTORS AND OUTER ARRAYS
Noise factors are those factors:
– that are not controllable.
– whose influences are not known.
– which are intentionally not controlled.
To determine robust design, experiments are conducted under the
influence of various noise factors.
An “Outer Array” is used to reduce the number of noise conditions
obtained by the combination of various noise factors.
For example:
Three 2-level noise factors can be combined using an L-4 into
four noise conditions(4 repetitions). Likewise seven 2-level
noise factors can be combined into eight conditions(8 repetitions)
using an L8 as an outer array.
When trial conditions are repeated without the formal “Outer Array” design, the noise conditions are considered random.
SCOPE AND SIZE OF EXPERIMENT
The scope of the study, i.e., cost and time availability, are factors that help determine the size of the experiment. The number of experiments that can be accomplished in a given period, and the associated costs are strictly dependent on the type of project under study. The total number of samples available divided by the number of repetitions yields the size of the array for design. The array size dictates the number of factors and their appropriate levels included in the study. Example: Several factors are identified for an optimization study. The time available is two weeks during which only 25 tests can be run. – Three repetitions for each trial condition are desired. – Array size 25/3 –> 8 L-8 array. – Seven of the identified 2-level factors can be studied.
ORDER OF RUNNING EXPERIMENTS
There are two common ways of running experiments. Suppose an experiment uses an L-8 array and each trial is repeated 3 times. How are the 3×8=24 experiments carried out? REPLICATION – The most desirable way is to run these 24 in random order. REPETITION – The most practical way may be to select the trial condition in random order and then complete all repetitions in that trial. NOTE: In developing conclusions from the results of designed experiments and assigning statistical significance, it is assumed that the experiments were unbiased in any way, thus randomness is desired and should be maintained when possible. MINIMUM REQUIREMENT – A minimum of one experiment per trial condition is required. Avoid running an experiment in an upward or downward sequence of trial numbers.
REPETITIONS AND REPLICATIONS
REPETITION: Repeat a trial condition of the experiment with/without a noise factor (outer array). Example: L-8 inner array and L-4 outer array. 8×4 = 32 samples. Select a trial condition randomly and complete all 4 samples. Take the next trial at random and continue. REPLICATION: Conduct all the trials and repetitions in a completely randomized order. In the above example, select one sample at a time in random order from among the 24 (8×4). NOTE: Results from replication contain more information than those from repetition. Since replication requires resetting the the same trial condition, it captures variation in results due to the experimental setup.
AVAILABLE ORTHOGONAL ARRAYS
The following Standard Orthogonal Arrays are commonly used to design experiments: 2-Level Arrays: L-4 L-8 L-12 L-16 L-32 L-64 3-Level Arrays: L-9 L-18 L-27 (L-18 has one 2-level column) 4-Level Arrays: L-16 & L-32 Modified
TRIANGULAR TABLE/LINEAR GRAPHS
TRIANGULAR TABLE OF INTERACTIONS (2-LEVEL COLUMNS) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (1) 3 2 5 4 7 6 9 8 11 10 13 12 15 14 (2) 1 6 7 4 5 10 11 8 9 14 15 12 13 (3) 7 6 5 4 11 10 9 8 15 14 13 12 (4) 1 2 3 12 13 14 15 8 9 10 11 (5) 3 2 13 12 15 14 9 8 11 10 (6) 1 14 15 12 13 10 11 8 9 (7) 15 14 13 12 11 10 9 8 (8) 1 2 3 4 5 6 7 (9) 3 2 5 4 7 6 (10) 1 6 7 4 5 (11) 7 6 5 4 (12) 1 2 3 (13) 3 2 (14) 1 (15) (Interaction tables for 3-level and 4-level factors are not shown here) LINEAR GRAPHS – Linear graphs are graphical representations of certain readings of the Triangular table for the convenience of experiment designs.The graphs consist of a combination of a line with circles/balls at the ends. The endpoints represent the columns where the interacting factors are assigned and the number associated with the line indicates the column number for the interaction. Example: For L-4 Orthogonal array, 1 x 2 => 3, which will be shown in graph form as 3 1 o————————-o 2 Complicated Linear Graphs for higher-order arrays are not shown here.
UPGRADING A COLUMN
COLUMN MODIFICATIONS: PREPARING A 4-LEVEL COLUMN – Select 3 2-level columns that are naturally interacting. Pick two and discard the third.Use the two columns to generate a new column. Follow these rules to combine the new columns: Old Columns New Column 1 1 ——-> 1 1 2 ——-> 2 2 1 ——-> 3 2 2 ——-> 4 Example: Suppose factor A is a 4-level factor. Using columns 1 2 3 of an L-16, a new 4-level column can be prepared and factor A assigned. PREPARING AN 8-LEVEL COLUMN – An 8-level column can be prepared from three of the seven interacting columns of an L-16. (Use columns 1 2 & 4, discard 3 5 6 & 7.) Follow these rules: Old Columns New Column ____________________________ 1 1 1 ——-> 1 1 1 2 ——-> 2 1 2 1 ——-> 3 1 2 2 ——-> 4 2 1 1 ——-> 5 2 1 2 ——-> 6 2 2 1 ——-> 7 2 2 2 ——-> 8 Note: An eight-level factor/column is not supported by QUALITEK-4 software. The above information is for user reference only.
DUMMY TREATMENT
This method allows a 3-level column to be made into a 2 -2-level column or a 4-level column into a 3-level column (e.g. levels 1 2 3 to 1 2 1′).The notation 1′ is used to keep track of the changed status only. For level assignment 1’=1. The selection of the level to be treated is arbitrary. Example: Three 3-level factors and one 2-level factor. – Use an L-9. Dummy treat any column and assign the 2-level factor.
RESULTS OF MULTIPLE CRITERIA
Frequently, your experiment may involve evaluating results in terms of more than one criterion of evaluation. For example, in a cake-baking experiment, the cakes baked under different recipes (trial conditions) may be evaluated by taste, looks, and moistness. These criteria may be subjective and objective. The best recipes can be determined by analyzing the results of each criterion separately. The recipes for optimum conditions determined this way may or may not be the same. Thus, it may be desirable to combine the evaluations under different criteria into one single overall criterion and use them for analysis. To combine readings under different evaluation criteria, they must first be normalized (unitless), and then combined with proper weighting. Furthermore, all evaluations must be of the same quality characteristic, i.e., either bigger or smaller is a better type. When an evaluation is of the opposite it can be subtracted from a larger constant to conform to the desired characteristic [(X2ref. – X2) instead of X2]. To combine all evaluations into a single criterion, Assume: X1 = Numeric evaluation under criterion 1 X1ref = Highest numerical value X1 can assume Wt1 = Relative weighting of criterion 1 Then an Overall Evaluation Criterion (OEC) can be defined as: OEC = (X1/X1ref)xWt1 + (X2/X2ref)xWt2 + …….