SYSEN 5300 Assignment 8 / Takehome Final Factorial Design at Two Levels and Response Surface Method
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Lec22DesignInnovationandDiscovery-FactorialDesignatTwoLevels.pptSYSEN 5300 (5310, 5320) - Systems Engineering and Six-Sigma for Systems Reliability and Quality
Introduction System Reliability (FMEA, Fault Tree) Six-sigma & Stat. Control Six-sigma & Systems Improvement (DOE) Six-sigma & Systems Improvement (RSM)
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SYSEN5300 Lecture 22 Design of Experiments: Factorial Designs at Two Levels
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H. Oliver Gao *
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Introduction
- Factorial design: select a fixed number of “levels” of each of a number of factors (variables), then run experiments in all possible combinations. We’ll discuss cases where there are just two levels of each factor
- Factors: quantitative (temperature, concentration); qualitative (two types of catalysts or presence and absence of some entity)
- Importance of two-level factorial designs (1. few runs per factor; 2. easy interpretation; 3. direct further experimentation; 4. designs can be suitably augmented; 5. basis for two-level fractional factorial designs for factor-screening)
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H. Oliver Gao *
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Example 1: Three Factors on Clarity of Film
- The cloudiness of a floor was affected by the amounts of emulsifiers A, B, and catalyst C, each at two levels, i.e., a 2 by 2 by 2 design.
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H. Oliver Gao *
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Example 2: Three Factors on Three Properties of Polymer Solution
- 3 factors: amount of reactive monomer; type of chain length regulator; amount of chain length regulator
- 3 responses, was the solution 1) milky (y1); 2) viscous (y2); and 3) yellow.
- Question: a product was required that was not milky, was of low viscosity, and had no yellowness, what shall we do?
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H. Oliver Gao *
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Which Factors Do What to Which Responses?
- In order to solve problems and make discoveries in science and engineering, it is very important to answer this question
- Factorial designs provide an economical way of doing this.
- The dimensional information of this kind can also suggest physical explanations for the effects, providing valuable directions for further inquiry
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H. Oliver Gao *
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Example 3: Quantitative Response in Pilot Plant Investigation
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H. Oliver Gao *
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Main Effects
- Main effect =
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H. Oliver Gao *
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Factorial Experiments vs. One-Factor-at-a-Time Method
- One-Factor-at-a-Time Method: factors are varied one at a time with the remaining factors held constant. It only provides an estimate of the effect of a single factor at selected and fixed conditions of the other factors
- For such an estimate to have more general relevance we have to assume the effect was the same at all the other setting--additivity
- Factorial design is able to detect and estimate interactions that measure nonadditivity.
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H. Oliver Gao *
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Interaction Effects
- Two-factor interactions: two factors have a coupled influence upon the response beyond their main effects (e.g., temperature and catalyst)
- An interaction can be measured by the difference between the temperature effects at the plus and minus levels of the catalyst factor.
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H. Oliver Gao *
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Interaction Effects (cont’d)
- Three-factor interactions: consider the TC interaction. Two measures of the TC interaction at the two K levels
- The difference between these two measures the consistency of the TC interaction for the two catalysts. Half of this difference is defined as the three-factor interaction of T, C, and K. TCK=(2-1)/2=0.5 (symmetric in all the factors)
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H. Oliver Gao *
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Summary of Effects
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H. Oliver Gao *
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Genuine Replicate Runs
- Randomization of run order for the all 16 runs ensures that the replication is genuine (i.e., variation between runs made at the same experimental conditions is a reflection of the total run-to-run variability).
- A genuine run replicate involves taking all the steps all over again.
Each estimated effect T, C, K, TC, … is a difference between two averages of 8 obs., the variance of effect
Var(Effect)=(1/8+1/8)s2
=8/4=2
SE(Effect)=1.4
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H. Oliver Gao *
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Estimate of the Error Variance and Standard Errors of the Effects
- In general, if each factor combination was replicated, a pooled estimate of the experimental run variance from g factor combinations:
- What is s2 in pilot plant example? What is variance (effect) and SE (effect)?
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H. Oliver Gao *
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Interpretation of Results
- It is important to determine which effects are almost certainly real and which might readily be explained by chance.
- A rough rule is that effects greater than 2-3 times their standard error are not easily explained by chance alone.
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H. Oliver Gao *
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Interpretation of Results (cont’d)
- Tentative conclusions:
- The effect of changing the concentration (C) over the ranges studied is to reduce the yield by about five units, and this is approximately so irrespective of the tested levels of the other variables
- The effects of temperature (T) and catalyst (K) can not be interpreted separately because of the large TK interaction.
- Very different behaviors of the two “catalyst types” in response to temperature. This was unexpected because although they were obtained from two suppliers, the catalysts were supposedly identical.
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H. Oliver Gao *
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Table of contrasts
- Orthogonality ensures that each estimated effect is unaffected by the magnitudes and signs of the others
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H. Oliver Gao *
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Exercise:
- For the following data calculate the main effects and interactions and their standard errors.
Answer: Factor 1=-2.2; Factor 2=2.5; Factor 3=-2.0; 1*2=-0.3; 1*3=-0.2; 2*3=0.2; 1*2*3=0.0; standard error of effect =0.3
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H. Oliver Gao *
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Dealing with more than one response
- Factorial experiments are of even greater value when a number of different responses y1, y2, .. are measured in each experiment run.
- Example: the manufacturer of pet food had received complaints that packages of food pellets received by customer contained too large amount of powder (y1). For a batch of product the value of y1 was not known but y2, the amount of powder produced in plant, could be measured easily. Control of the plant had been attempted by keeping y2 low.
- Effect of changing manufacturing conditions: temperature (A), flow (B), and zone (C). A total of just 8 runs were made.
- In each run, four responses were measured: powder in product y1, powder in plant y2, yield y3, and energy consumption y4.
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H. Oliver Gao *
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Dealing with more than one response (Cont’d)
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H. Oliver Gao *
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Dealing with more than one response (Cont’d)
- questions and conclusions.
- Ideally, the experimenter would like to decrease the powder y1, increase yield y3, and decrease energy y4. How? compromise
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H. Oliver Gao *
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Dealing with more than one response –Eyeball contour plots
- Assuming 0.5 cents savings per unit of increase in yield y3, and 0.2 cents costs per unit of increase in energy y4
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H. Oliver Gao *
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Dealing with more than one response –Surprise and question
- The powder in the plant y2 previously used to control the process was not related to y1, or, indeed, to any of the factors. Why?
- Increasing temperature with low flow rate would have been expected to increase energy consumption but actually reduced it. Why?
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H. Oliver Gao *
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A 24 Factorial Design
- Often there are more factors to be investigated than can conveniently be accommodated within the time and budget available. Usually we can separate genuine effects from noise without replication.
- E.g., a process development experiment with 4 factors: amount of catalyst charge 1, temperature 2, pressure 3, and concentration of a reactant 4
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H. Oliver Gao *
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A 24 Factorial Design (cont’d)
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H. Oliver Gao *
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A 24 Factorial Design (cont’d)
No direct estimate of the experimental error variance is available from this 16-run since there are no replicates.
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H. Oliver Gao *
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A 24 Factorial Design-Interpretation
- A increase in catalyst charge (factor 1) from 10 to 15 reduces conversion by 8%, and the effect is consistent over the levels of the other factors.
- The effects of factors 2 (temperature) and 4 (concentration) need to be considered jointly.
- Factor 3 is essentially inert.
- The corresponding pairs of data at the two levels of factor 3 should act approximately as replicates.
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