 # DOE Process Optimization

Oct 24, 2014

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#### 2k factorial

A + + + + -z

B + + + + 0

C + + + +

D z

2

Design of Experiments for Process OptimizationA 2 days Practical WorkshopHo : 1 = 2 ; Ha : 1 2 * SSTotal = SSFactors + SSErrors * I=ABCDE ; A=BCDE ; AB = CDE * x-(z s/n) 2 2. Calculate the appropriate statistics. 3. Test hypothesis (at 95% or 99% confidence).

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Design of Experiments for Process Optimization

Statistical deduction for 2 distributions 1. 2 distributions can be compared for the differences in the means as well as the differences in the variances. 2. Normally we can compare the sum of squares of 2 distributions if they are from normal distribution family. 3. The test statistics to be used here will be F-Statistics.

Sum of Squares due to effect A Sum of Squares due to effect B where Example : Factors Process Random Errors/Noises Output data distribution:

~

Fdf A, df B,

df A - degree of freedom for A df B - degree of freedom for B

Output

The variation of the output can be stratified into variations due to random errors and also variations due to factors. This can be represented as: SSTotal = SSFactors + SSErrors where SS is the Sum of Squares. Application examples will be shown in Factorial Designs.

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Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

One-Sample Hypothesis Testing Procedure Objective: To test on the basis of sampled data whether the population mean or population percentage differs from specified standard or historical value.

One-Sample Hypothesis Test of Means Case 1: Example: In the past, mean library usage per cardholder was 8.5 books during the year. A random sample of 100 cardholders showed the following results this year: x = 9.34 books, s = 3.31 books. The library administration would like to know whether this years mean usage () has changed from that for past years. Conduct the appropriate test, controlling the risk at .05. State the alternatives. n 30 and unknown

Steps in the Hypothesis Testing Procedure: 1. Ho : = 8.5 Sample data is used to decide whether or not Ho is rejected.

2. H1 : 8.5 H1 : > 8.5 H1 : < 8.5

two-sided alternative one-sided (upper-tail) alternative one-sided (lower-tail) alternative

Compare mean of sampled population ( x = 9.34) to the specified standard or historical value of the population mean (o = 8.5). Select alternative hypothesis depending on the nature of problem at hand.

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Design of Experiments for Process Optimization

3. Level of significance of the test, = .05. A 5% risk of erroneously rejecting Ho when Ho is true. In practice, = .05 or .01. 4. Since n is greater than 30, use the z-distribution. Test Statistic: Z =x - o

s/ n

=

9.34 8.5 (3.31/ 100)

= 2.54

Note: If is known, replace s with . 5. Acceptance and Critical regions with = .05 Two-sided alternative:

Critical Region

Acceptance Region (do not reject Ho)

Critical Region (Reject Ho)

z -z.025 = -1.96 0 z.025 = 1.96

The values 1.96 and -1.96 are included in the acceptance region. If Ho is rejected, we conclude that the true population mean does not equal to 8.5. It does not matter if the population mean is more or less than 8.5.

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Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

One-sided (upper-tail) alternative: Critical Region

Acceptance

Region

z 0 z.05 = 1.645

Ho will be rejected if the value of the sample mean is significantly higher than 8.5 . One-sided (lower-tail) alternative:

Critical Region

Acceptance Region

z -z.05 = -1.645 0

Ho will be rejected if the value of the sample mean is significantly lower than 8.5 . 6. Conclusion: For : H1 : > 8.5 , reject Ho : = 8.5 since 2.54 is greater than 1.645 .

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Design of Experiments for Process Optimization

Note: Acceptance and Critical Regions with = 0.01 Two-sided alternative:

Critical Region Acceptance Region

Critical Region

z -z.005 = -2.58 0 z.005 = 2.58

One-sided (upper-tail) alternative: Critical Region Acceptance Region

z 0 z.01 = 2.33

One-sided (lower-tail) alternative: Critical Region

Acceptance Region

z -z.01 = -2.33 019

Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

Hypothesis:

Ho : = o HA : o

Test statistic: Z = x - o /n or x - o s/n If n 30 and unknown

Rejection region:

Reject Ho if z < -z/2 or z > z/2

Reject

Reject

/2

/2 z

-z/2

0

z/2

Hypothesis:

Ho : o HA : > o

Test statistic: Z = x - o /n or x - o s/n If n 30 and unknown

Rejection region:

Reject Ho if z > z Reject

z 0

z

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Design of Experiments for Process Optimization

Hypothesis:

Ho : o HA : < o

Test statistic: Z = x - o /n or x - o s/n If n 30 and unknown

Rejection region:

Reject Ho if z < -z

Reject

z

-z

0

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Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

One-Sample Hypothesis Test of Percentages (n 100) Example 1: One-sided (lower-tail) alternative

The manager of Company A advertised that 90 percent of its customers are satisfied with the companys services. The manager of Company B feels that this is an exaggerated statement. In a random sample of n = 150 of Company As clients, 132 (x) said they were satisfied. What should be concluded if a test were conducted at the .05 level of significance? Solution: Hypothesis Testing Procedure is essentially the same procedure used for testing means with a large sample size except for the following: Test Statistic: z p = = ( P o ) / p ( x/n ) 100 Hypothesized value of the population percentage Standard deviation of percentage

o = p == Ho H1 p : :

o ( 100 - o ) / n

= 90% ( o = 90%) < 90%

= .05 = 132 / 150 x 100 = 90 ( 100 90 ) / 150 = 88% = 2.4% = -.833

p

z = ( 88 90 ) / 2.4

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Design of Experiments for Process Optimization

Example 2:

One-sided (upper-tail) alternative

A company anticipates that 5% of its employees drive to work. A random sample of 200 employees showed that 9% (p) drive to work. Using a significance level of .01, conduct a statistical test to test the accuracy of the companys assumption. Solution: Ho H1 : : = = =

= 5% > 5%.01 5 ( 100 5 ) / 200 ( 9 - 5 ) / 1.5 = 1.5% = 2.67

pz

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Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

Example 3: Suppose the target property value in the previous example is 120. How may one determine whether the production process has deviated from this target?

Outline of solution: We may answer this question by a test of hypothesis. Ho H1 Set: :

= 120 120 = 0.05

Test statistic:

z =

x - o /n

Critical region: Value of test statistic:

z < -1.96, z > 1.96 z= = 123.8 - 120 2/4 3.8

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Design of Experiments for Process Optimization

Robust Design for a Process

Robust Design for a Process

Coding of Factors It is very common for us to code the Process Factors. Normally we do not need to write the actual factor in the experimental array, we usually code them as Low (-), Medium (0) or High (+) with the following transformation. For example, say we have factor A, we denote AMin Mean of A Mean of A AMax Mean of A Mean of A AMedium Mean of A Mean of A

= -1 = +1 =0

Effects/Types of Factors There are three types of factors in general: 1. Critical factors factors that have direct and big impact to the process, usually there are only one or two factors of this type 2. Trivial factors factors that have little effects to the process. There are usually many factors of this type. 3. Dummy factors factors that have no effect at all to the process.

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Section 1: Introduction to concepts of Statistical variation, location and dispersion effects

Process Models Many models appear for production process. There may be Linear Effect Models like those that follow Ohms Law, Quadratic Models, Exponential Models or other non-linear models.

Generally we can represent a process model by Y = F(X) + Random Errors

The function F needs to be determined with Design of Experiments or Regression Analysis. In either case, data needs to be collected.

Robust Designs A robust design will meet the following criteria:

1. The parameters combination gives the optimum performance. 2. The process is NOT sensitive to errors or noises. 3. Critical parameters have been determined and controlled appropriately. 4. Output of the process is not varied greatly by any small variations in the input parameters or materials.

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Design of Experiments for Process Optimization

Selection of Factors Factors considered for experimentation can be selected from: 1. Engineering knowledge. 2. Experiences. 3. From previous experiments.

Golden rules for experimentation: 1. Start with large number of factors at the initial experiment. 2. The range of factors should be set as close to the limit as possible.

Funneling of factors during experimentation: Stage of Experiments Initial Experiment Number of Factors Many Factors Critical Factors (from initial experiment) Optimize

Secondary Experiment

Fine Tuning Experiment

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