MATH602: APPLIED STATISTICS

Size: px
Start display at page:

Download "MATH602: APPLIED STATISTICS"

Transcription

1 MATH602: APPLIED STATISTICS Dr. Srinivas R. Chakravarthy Department of Science and Mathematics KETTERING UNIVERSITY Flint, MI Lecture 10 1

2 FRACTIONAL FACTORIAL DESIGNS Complete factorial designs cannot always be conducted. Why? 7 factors at 2 levels requires 128 runs, a significantly large number. In some cases, it impossible to run all these 128 combinations. In some cases, the commercial production may have to be stopped for the duration of the study. 2

3 In any case, the expenses related to such experiments are prohibitively large. Often higher order interactions are insignificant. no distinguishable effects are noticed when moderately large number of factors are used. Fractional factorial designs (FFD) are significant alternatives to the above situations. The fraction is a carefully selected subset of all combinations. 3

4 In many cases, the experiments that involve many factors are routinely conducted as fractional factorials to identify factor-level combinations for a future full study using complete factorial designs. It is important to note that 1. the analysis of a FFD is relatively simple and straightforward; and 2. the use of a FFD doesn t stop us from completing a full factorial design. 4

5 CONFOUNDING IN FFD In a complete factorial design involving k factors at two levels, we have 2 k units, which and the analysis of such a design results in estimating k main effects and (2 k -k-1) interaction effects. Suppose we look at a fractional factorial design, say, the fraction 1/2 p. That is, there will be 2 k-p experimental units. Obviously, some effects are going to be confounded with one another. 5

6 What is confounding? Two or more effects are said to be confounded if calculated effects can only be attributed to their joint influence on the response, and not to their individual ones. That is to say that the contrasts of the effects will be the same (except possibly for the sign). 6

7 Construction and Analysis of FFD Here we will illustrate how to construct and analyze fractional factorial designs through a number of examples. Defining Relation: FFD s are generated using one or more generators. The relation that defines the generator is called the defining relation. This is the key to the confounding pattern. 7

8 Analysis: The analysis of a FFD is very similar to what we saw earlier in the case of a factorial design. However, here one has to be even more careful in interpreting the results as some factor effects are confounded with one another. 8

9 DESIGN RESOLUTION In practice, an important tool that will be used in selecting a FFD out of many possible ones, is the concept of design resolution. This identifies the order of confounding of the main effects and the interactions. Design Resolution: A design is said to be a design of resolution R if no p-factor effect is confounded with any other effect containing less than R-p factors. 9

10 In general, the resolution of a 2-level FFD is the length of the shortest word in the defining relation. Note: The resolution of a design is denoted by appropriate Roman letter appended as a subscript. For example, R III refers to a design of resolution R = 3. Also, another notation that is commonly used to denote a FFD is 2 III k-p. This a p-fraction 2-level, k-factor FFD of resolution III. 10

11 To further grasp the idea of design resolution, let us look at the following examples: (a) A design resolution of III does not confound main effects with one another but does confound the main effects with two-factor interactions. (b) A design resolution of IV does not confound main effects and two-factor interactions but does confound the two-factor interactions with the other two-factor interactions. 11

12 (c) A design resolution of V does not confound main effects and two-factor interactions but does confound the two-factor interactions with the other three-factor interactions, and so on. HOW TO SELECT A FFD? Having seen how to construct a FFD using appropriate generator(s), and also know how to analyze a FFD, the question now is how to select a generator in practice? 12

13 That is, how to select an appropriate FFD for a problem under study in practice? Recall that no matter how best you choose a statistical technique for analysis, it will not do much good if the design is not properly chosen. Hence, it is imperative that one is able to identify a proper design for the problem under study. After identifying the important factors that influence the response variable for the problem under study, use 13

14 (a) the existing tables to choose a FFD (b) computer to generate a FFD (c) trial and error method to identify a FFD. 14

15 EXAMPLE 5: (2 4-1 design - Problem 7.18): A chemical product is produced in a pressure vessel Four factors at two levels each A Temperature B Pressure C Concentration of Formaldehyde D Stirring Rate Response is the filtration rate. 15

16 EXAMPLE 6: (2 6-2 design - Example 7.9): Shrinkage (Y) of parts in injection molding process. Six factors at two levels each A Molding Temperature B Screw Speed C Holding Time D Cycle Time E Gate Size F Holding Pressure 16

17 Use E = ABC and F = BCD as generators Defining relation is: I = ABCE = BCDF = ADEF Alias Structure : I + ABCE + ADEF + BCDF A + BCE + DEF + ABCDF B + ACE + CDF + ABDEF C + ABE + BDF + ACDEF D + AEF + BCF + ABCDE E + ABC + ADF + BCDEF F + ADE + BCD + ABCEF ABD + ACF + BEF + CDE ABF + ACD + BDE + CEF AB + CE + ACDF + BDEF AC + BE + ABDF + CDEF AD + EF + ABCF + BCDE AE + BC + DF + ABCDEF AF + DE + ABCD + BCEF BD + CF + ABEF + ACDE BF + CD + ABDE + ACEF 17

18 CONSTRUCTION OF A FFD A 2 k-1 FFD of highest resolution construction. Write down a full FD for k-1 factors. Add the k-th factor by identifying its high and low levels to those of the highest order-interaction. [Note: We can use any order interaction to assign the k-th factor, but will not get the highest resolution]. 18

19 A 2 k-2 FFD is constructed as follows Write down a full FD with k-2 factors. Choose two generators, say, P and Q. Design Generators: E = ABC and F = BCD 19

20 RESPONSE SURFACE METHODOLOGY RSM is collection of statistical and mathematical methods that are used in modeling a response variable, as a function of 2 or more predictor variables and the objective is to find optimum settings that will optimize the response variable. Suppose the proposed model (when k=2) is of the form: Y = f(x 1, X 2 ) + e or equivalently E(Y) = f(x 1, X 2 ). The surface represented by the graph of E(Y) = f(x 1,X 2 ) is known as response 20

21 surface. The projection of this surface for fixed values of E(Y) onto the (X 1,X 2 )-plane is referred to as the contour plot. Fitting RSM models using MINITAB is very similar to the Factorial designs. In addition to these, there are other options that are unique to RSM. These include surface and contour plots and searching for optimum settings for the parameters. 21

22 Most applications of RSM are sequential in nature. Start with a screening experiment (reduce a long list of variables into a smaller set) Determine whether the current levels of the factors under study result in optimum for the response value. If not, optimize using method of steepest ascent or descent. When the process is near optimum, obtain a reasonable function that is quite descriptive of the real situation around the region of optimum. Usually a second-order model will be needed. 22

23 Imagine a 2 2 CENTRAL COMPOSITE DESIGNS design in which first-order model is not appropriate. To introduce quadratic terms and estimate the parameters require running additional experiments at selected points. CCD is a simple and highly efficient design to the above problem. CCD is performed often in a sequential setting. First we run a FD (or a FFD) and if there is evidence of lack of fit, we augment this with axial runs. 23

24 To generate a CCD we must specify a, the distance of the axial runs from the center of the design, and n C, the number of center points. Usually 3 n C 5. How do we choose a? This is chosen so that the response surface design is rotatable. By a rotatable design, we mean a design in which the variance of the response, s 2 x' (X'X) -1 x, is the same at all points x that have the same distance from the center of the design. In a 2 k design a = (2 k ) Thus, if k = 2, a = 2. If k = 4, then a = 2. 24

25 25 ILLUSTRATIVE EXAMPLE

26 TAGUCHI APPROACH TO QUALITY So far we saw the classical DOE and some specific designs, and their analysis. Dr. Genichi Taguchi of Japan has incorporated a number of quality engineering methods that use DOE with an idea to design high-quality systems at reduced cost. These methods provide an efficient and systematic approach to optimize designs for performance, quality and cost. These methods emphasize designing quality into the products and processes, compared to the standard method of inspection for quality. 26

27 These have been very effective in improving quality of Japanese products, and hence got popular in western industries. Although there is some controversy about Taguchis methods in terms of the philosophical and technical aspects, here we will briefly illustrate the basic ideas and the usefulness of these designs in practice. Note that until before Taguchi's methods were introduced, traditional designs were used only to assess effect on averages. However, Taguchi made experimenters aware of the value in using the designs to assess the impact of factors on the variability of the response variable. 27

28 In practice, one usually deals with many factors to study a response variable. This leads to the study of many test combinations in the case of a full study. A standard method of reducing the number of test runs is to use fractional factorial designs (recall this from earlier lecture on DOE). Taguchi constructed a special set of general designs that consist of orthogonal arrays. These determine the least number of test runs for a given number of factors to be studied. 28

29 Taguchi's loss function L(y) is: L(y) = k (y - y 0 ) 2, where y represents the quality characteristic (such as dimension, speed, rate, performance), y 0 is the target value for y, and k is a constant that is dependent on the cost structure of the manufacturing process that produces the product. This function possesses the following properties: (a) the loss must be zero when target is attained; (b) the magnitude of the loss increases rapidly as y deviates farther away from the target value; (c) the loss function is a continuous function of the deviation 29

30 No matter how the quality of a product is defined, the measure will fall under one of the following three characteristics: (A) the smaller the better (B) the larger the better (C) the nominal is best 30

31 ORTHOGONAL ARRAYS Earlier we saw how factorial designs (FD) and fractional factorial designs (FFD) were used in practice. FFD is a means to reduce the number of runs needed when the number of factors is large. In choosing a FFD for a study, certain treatment conditions are chosen in order to maintain the orthogonality among the various main effects and interaction effects. Orthogonal arrays were first recorded sometime in 1897 by French Mathematician J. Hadamard; but the utility of these 31

32 were not explored until World War II by British Statisticians: Plackett and Burman. Taguchi has developed a family of FFD's that can be utilized in various situations. The associated design matrices are labeled as L * where * is a selected positive integer. L stands for Latin square design, as these designs are simply a form of well known designs such as Plackett-Burman, FFD or Latin square designs. However, the combination of the loss function and the robust designs to find optimal settings is one of the strongest aspects of Taguchi's method. 32

33 The construction and the use of Latin squares orthogonal arrays date back to the period of World War II mainly in the context of agricultural applications. Since then they have been used extensively in many applications. Taguchi constructed a new set of OA's using the orthogonal Latin squares in a unique way. This construction along with a set of rules for choosing a particular OA has simplified the task for many engineers and statisticians, who apply DOE in practice. Below we will illustrate the use of Taguchi's OA by taking L 4 and L 8 OA s. 33

34 L 4 Orthogonal Array Table 1: L 4 Orthogonal Array Run Column number

35 From the above table, we see that an L 4 experiment consists of four rows and three columns. Each row corresponds to a particular run in the experiment and each column corresponds to the factors specified in the study. Each column contains 2 low levels and 2 high levels for the factor assigned to that column. We use -1 for low level and 1 for high level. In the first run, for example, the three design variables are set at their low level and in the second run, the first parameter is set at low level and the remaining two variables are set at high level, and so on. 35

36 L 8 Orthogonal Array Table 2: L 8 Orthogonal Array Run Column number

37 Notice that this is a one-sixteenth FFD which has only the 8 of the total possible 128 runs. The seven columns are used to assign up to 7 factors. If only three factors are of interest, then this will become a full factorial design. When all columns are assigned a factor, this is known as a saturated design. In general, a FFD is said to be saturated when the design only allows for the estimation of main effects. These designs are effective when used as part of screening process when there are a number of factors to be examined. 37

38 There are two sets of Taguchi's OA's available for use in practice: one set deals with 2-level factors, which are denoted by L 4, L 8, L 12, L 16, L 32 ; the second set deals with 3- level factors: L 9, L 18, and L 27. SELECTION OF AN OA: The selection of an OA is easily achieved once the number of degrees of freedom in the study is determined along with the number of levels of the factors. Taguchi has tabulated a total of 18 standard orthogonal arrays which can be found in many text books dealing with Taguchi methods (see some references at the end of the handout). 38

39 ASSIGNMENT OF FACTORS TO COLUMNS: As you would have seen there are several columns available in an OA. The question is how to assign the factors (and the interactions of the factors, if any) to these columns? They cannot be done arbitrarily as confounding of the factors will result. If there are no significant interaction effects among the factors, then arbitrarily we can assign factors to columns. Taguchi devised a scheme to assign factors and interactions using linear graphs and triangular tables. The purpose of linear graphs is to indicate which factors may be assigned to which columns and the purpose of the 39

40 triangular tables is to identify appropriate columns to assign the interactions. Usually there will be more than one choice for the assignment. However, careful assignment is necessary to avoid any unnecessary confounding effects. Any unassigned columns will be used to estimate the error sum of squares. 40

41 Example 1: Consider an L 4 OA. This one has 3 columns and so three factors or two factors and an interaction of these two factors can be assigned to these columns. For example, Factor A can be assigned to column 1 and Factor B can be assigned to column 2. The interaction AB is now assigned to column 3. Now this assignment resembles the design matrix corresponding to a 2 2 full factorial design except that the interaction effect is really -AB.[Why?]. However, if interaction term is negligible or if one is not interested in it, then another factor, say, Factor C can be assigned to column 3. 41

42 Example 2: Consider an L 8 OA. This one has 7 columns and so 4 to 7 factors or a full factorial design with 3 factors at two levels can be studied. For studying a 2 3 full factorial design, we can assign Factors A, B and C to columns 1, 2 and 4, respectively. Note that we cannot assign these three factors to the first three columns as this will result in confounding the effects of Factor C and the interaction of AB. One has to use the linear graphs or the triangular tables to appropriately assign the factors. Also, this OA can be used to screen 4 to 7 factors, by assigning them properly. 42

43 INNER AND OUTER ARRAYS Taguchi proposed a collection of techniques to identify the settings for the controlled factors that will yield a robust performance. These include the selection of the DOE and the statistical analysis of the data. First select a DOE for the controlled factors. The design matrix for this is referred to as an inner array. Now select a DOE for the noise factors. The design matrix for this is referred to as an outer array. For each combination of factors in an inner array, run all combinations of the noise factors in the outer array. 43

44 Taguchi classifies the parameter design problems into different categories and the effects are evaluated using the concept: signal-to-noise ratio (A) The smaller the better: The target value for the response is zero (why?). Thus, for this S/N ratio= - 10 log( i n = 1 y 2 i / n The goal of the experiment here is to minimize the sum of the squared response values, which is equivalent to maximizing the S/N ratio. ) 44

45 (B) The larger the better: Here the goal is to maximize the response variable, which is equivalent to minimizing the reciprocal of the response value. Thus, S/N ratio= - 10 n 1 log 2 i = 1 nyi The goal of the experiment here is to maximize the S/N ratio. 45

46 (C) The nominal the better: For this case Taguchi recommends using: where s 2 2 y S/N ratio= 10 log 2 s is the sample variance of the observations. 46

47 ANALYSIS OF OA DESIGNS The analysis of designs based on OAs is very similar to the analysis of FFDs seen earlier. But now there is some interest in the effect of the variation also. Using S/N ratio, the effect of variation is studied. G.E.P. Box (Signal-to-Noise ratios, performance criteria, and transformations, Technometrics1988) states that the use of S/N ratio concept is equivalent to an analysis of the logarithm of the data. This is due to the fact that the assumption of the variance is proportional to the mean requires a logarithmic transformation to stabilize the condition. 47

48 ILLUSTRATIVE EXAMPLE 1 Run A B C y 1 y 2 y s

49 ILLUSTRATIVE EXAMPLE 2 An injection molding process engineer is interested in identifying the factors that contribute to variability in part shrinkage as well as to determine the best settings for the factors that will minimize the shrinkage. Part shrinkage is measured as the amount of deviation from the desired part size. Seven controlled factors A-Cycle time; B-Mold temperature; C-Holding pressure; D-Gate size; E-Cavity thickness; F-Holding time; G-Screw speed; and 3 noise factors: H-Ambient temperature; I-Moisture content; J- Percent regrind; were identified after a brain storming session. The data for this study is given in the following table. 49

50 L 8 : OA for Illustrative Example 2 Run A B C D E F G Y 1 Y 2 Y 3 Y

The One-Quarter Fraction

The One-Quarter Fraction The One-Quarter Fraction ST 516 Need two generating relations. E.g. a 2 6 2 design, with generating relations I = ABCE and I = BCDF. Product of these is ADEF. Complete defining relation is I = ABCE = BCDF

More information

Fractional Replications

Fractional Replications Chapter 11 Fractional Replications Consider the set up of complete factorial experiment, say k. If there are four factors, then the total number of plots needed to conduct the experiment is 4 = 1. When

More information

Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing

Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing Vladimir Brayman Webtrends October 19, 2012 Advantages of Conducting Designed Experiments in Digital Marketing Availability

More information

MATH602: APPLIED STATISTICS Winter 2000

MATH602: APPLIED STATISTICS Winter 2000 MATH602: APPLIED STATISTICS Winter 2000 Dr. Srinivas R. Chakravarthy Department of Industrial and Manufacturing Engineering & Business Kettering University (Formerly GMI Engineering & Management Institute)

More information

FRACTIONAL FACTORIAL

FRACTIONAL FACTORIAL FRACTIONAL FACTORIAL NURNABI MEHERUL ALAM M.Sc. (Agricultural Statistics), Roll No. 443 I.A.S.R.I, Library Avenue, New Delhi- Chairperson: Dr. P.K. Batra Abstract: Fractional replication can be defined

More information

Reference: Chapter 8 of Montgomery (8e)

Reference: Chapter 8 of Montgomery (8e) Reference: Chapter 8 of Montgomery (8e) 69 Maghsoodloo Fractional Factorials (or Replicates) For Base 2 Designs As the number of factors in a 2 k factorial experiment increases, the number of runs (or

More information

FRACTIONAL REPLICATION

FRACTIONAL REPLICATION FRACTIONAL REPLICATION M.L.Agarwal Department of Statistics, University of Delhi, Delhi -. In a factorial experiment, when the number of treatment combinations is very large, it will be beyond the resources

More information

Lecture 12: 2 k p Fractional Factorial Design

Lecture 12: 2 k p Fractional Factorial Design Lecture 12: 2 k p Fractional Factorial Design Montgomery: Chapter 8 Page 1 Fundamental Principles Regarding Factorial Effects Suppose there are k factors (A,B,...,J,K) in an experiment. All possible factorial

More information

On the Compounds of Hat Matrix for Six-Factor Central Composite Design with Fractional Replicates of the Factorial Portion

On the Compounds of Hat Matrix for Six-Factor Central Composite Design with Fractional Replicates of the Factorial Portion American Journal of Computational and Applied Mathematics 017, 7(4): 95-114 DOI: 10.593/j.ajcam.0170704.0 On the Compounds of Hat Matrix for Six-Factor Central Composite Design with Fractional Replicates

More information

Minimum Aberration and Related Criteria for Fractional Factorial Designs

Minimum Aberration and Related Criteria for Fractional Factorial Designs Minimum Aberration and Related Criteria for Fractional Factorial Designs Hegang Chen Division of Biostatistics and Bioinformatics 660 West Redwood Street University of Maryland School of Medicine Baltimore,

More information

A Survey of Rational Diophantine Sextuples of Low Height

A Survey of Rational Diophantine Sextuples of Low Height A Survey of Rational Diophantine Sextuples of Low Height Philip E Gibbs philegibbs@gmail.com A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any

More information

TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS

TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS STAT 512 2-Level Factorial Experiments: Regular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS Bottom Line: A regular fractional factorial design consists of the treatments

More information

A UNIFIED APPROACH TO FACTORIAL DESIGNS WITH RANDOMIZATION RESTRICTIONS

A UNIFIED APPROACH TO FACTORIAL DESIGNS WITH RANDOMIZATION RESTRICTIONS Calcutta Statistical Association Bulletin Vol. 65 (Special 8th Triennial Symposium Proceedings Volume) 2013, Nos. 257-260 A UNIFIED APPROACH TO FACTORIAL DESIGNS WITH RANDOMIZATION RESTRICTIONS PRITAM

More information

Probability Distribution

Probability Distribution Probability Distribution 1. In scenario 2, the particle size distribution from the mill is: Counts 81

More information

Lecture 14: 2 k p Fractional Factorial Design

Lecture 14: 2 k p Fractional Factorial Design Lecture 14: 2 k p Fractional Factorial Design Montgomery: Chapter 8 1 Lecture 14 Page 1 Fundamental Principles Regarding Factorial Effects Suppose there arek factors (A,B,...,J,K) in an experiment. All

More information

Solutions to Exercises

Solutions to Exercises 1 c Atkinson et al 2007, Optimum Experimental Designs, with SAS Solutions to Exercises 1. and 2. Certainly, the solutions to these questions will be different for every reader. Examples of the techniques

More information

Session 3 Fractional Factorial Designs 4

Session 3 Fractional Factorial Designs 4 Session 3 Fractional Factorial Designs 3 a Modification of a Bearing Example 3. Fractional Factorial Designs Two-level fractional factorial designs Confounding Blocking Two-Level Eight Run Orthogonal Array

More information

3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value.

3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. 3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. One-way ANOVA Source DF SS MS F P Factor 3 36.15??? Error??? Total 19 196.04 Completed table is: One-way

More information

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13:

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13: 1.0 ial Experiment Design by Block... 3 1.1 ial Experiment in Incomplete Block... 3 1. ial Experiment with Two Blocks... 3 1.3 ial Experiment with Four Blocks... 5 Example 1... 6.0 Fractional ial Experiment....1

More information

Design of Experiments (DOE) A Valuable Multi-Purpose Methodology

Design of Experiments (DOE) A Valuable Multi-Purpose Methodology Applied Mathematics, 2014, 5, 2120-2129 Published Online July 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.514206 Design of Experiments (DOE) A Valuable Multi-Purpose

More information

TWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS

TWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS STAT 512 2-Level Factorial Experiments: Irregular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS A major practical weakness of regular fractional factorial designs is that N must be a

More information

Statistica Sinica Preprint No: SS R1

Statistica Sinica Preprint No: SS R1 Statistica Sinica Preprint No: SS-2015-0161R1 Title Generators for Nonregular $2^{k-p}$ Designs Manuscript ID SS-2015-0161R1 URL http://www.stat.sinica.edu.tw/statistica/ DOI 10.5705/ss.202015.0161 Complete

More information

Design and Analysis of Experiments

Design and Analysis of Experiments Design and Analysis of Experiments Part VII: Fractional Factorial Designs Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com 2 k : increasing k the number of runs required

More information

Design of Experiments SUTD - 21/4/2015 1

Design of Experiments SUTD - 21/4/2015 1 Design of Experiments SUTD - 21/4/2015 1 Outline 1. Introduction 2. 2 k Factorial Design Exercise 3. Choice of Sample Size Exercise 4. 2 k p Fractional Factorial Design Exercise 5. Follow-up experimentation

More information

ST3232: Design and Analysis of Experiments

ST3232: Design and Analysis of Experiments Department of Statistics & Applied Probability 2:00-4:00 pm, Monday, April 8, 2013 Lecture 21: Fractional 2 p factorial designs The general principles A full 2 p factorial experiment might not be efficient

More information

Experimental design (DOE) - Design

Experimental design (DOE) - Design Experimental design (DOE) - Design Menu: QCExpert Experimental Design Design Full Factorial Fract Factorial This module designs a two-level multifactorial orthogonal plan 2 n k and perform its analysis.

More information

Fractional Factorials

Fractional Factorials Fractional Factorials Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 26 1 Fractional Factorials Number of runs required for full factorial grows quickly A 2 7 design requires 128

More information

Strategy of Experimentation III

Strategy of Experimentation III LECTURE 3 Strategy of Experimentation III Comments: Homework 1. Design Resolution A design is of resolution R if no p factor effect is confounded with any other effect containing less than R p factors.

More information

23. Fractional factorials - introduction

23. Fractional factorials - introduction 173 3. Fractional factorials - introduction Consider a 5 factorial. Even without replicates, there are 5 = 3 obs ns required to estimate the effects - 5 main effects, 10 two factor interactions, 10 three

More information

Design of Experiments SUTD 06/04/2016 1

Design of Experiments SUTD 06/04/2016 1 Design of Experiments SUTD 06/04/2016 1 Outline 1. Introduction 2. 2 k Factorial Design 3. Choice of Sample Size 4. 2 k p Fractional Factorial Design 5. Follow-up experimentation (folding over) with factorial

More information

Fractional Replication of The 2 k Design

Fractional Replication of The 2 k Design Fractional Replication of The 2 k Design Experiments with many factors involve a large number of possible treatments, even when all factors are used at only two levels. Often the available resources are

More information

A Comparison of Factor Based Methods for Analysing Some Non-regular Designs

A Comparison of Factor Based Methods for Analysing Some Non-regular Designs A Comparison of Factor Based Methods for Analysing Some Non-regular Designs Ebenezer Afrifa-Yamoah Master of Science in Mathematics (for international students) Submission date: June 2016 Supervisor: John

More information

Design of Engineering Experiments Chapter 8 The 2 k-p Fractional Factorial Design

Design of Engineering Experiments Chapter 8 The 2 k-p Fractional Factorial Design Design of Engineering Experiments Chapter 8 The 2 k-p Fractional Factorial Design Text reference, Chapter 8 Motivation for fractional factorials is obvious; as the number of factors becomes large enough

More information

Homework Assignments Sheet. 4) Symbol * beside a question means that a calculator may be used for that question. Chapter 1 Number 9 days

Homework Assignments Sheet. 4) Symbol * beside a question means that a calculator may be used for that question. Chapter 1 Number 9 days Riverside Secondary School Math 10: Foundations and Precalculus Homework Assignments Sheet Note: 1) WS stands for worksheet that will be handed out in class 2) Page numbers refer to the pages in the Workbook

More information

STAT451/551 Homework#11 Due: April 22, 2014

STAT451/551 Homework#11 Due: April 22, 2014 STAT451/551 Homework#11 Due: April 22, 2014 1. Read Chapter 8.3 8.9. 2. 8.4. SAS code is provided. 3. 8.18. 4. 8.24. 5. 8.45. 376 Chapter 8 Two-Level Fractional Factorial Designs more detail. Sequential

More information

β x λ Chapter 11. Supplemental Text Material

β x λ Chapter 11. Supplemental Text Material Chapter. Supplemental Text Material -. The Method of Steepest Ascent The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder model y = β + β x 0 and we wish to use

More information

TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING. Upper-case letters are associated with factors, or regressors of factorial effects, e.g.

TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING. Upper-case letters are associated with factors, or regressors of factorial effects, e.g. STAT 512 2-Level Factorial Experiments: Blocking 1 TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING Some Traditional Notation: Upper-case letters are associated with factors, or regressors of factorial effects,

More information

Design and Analysis of Multi-Factored Experiments

Design and Analysis of Multi-Factored Experiments Design and Analysis of Multi-Factored Experiments Fractional Factorial Designs L. M. Lye DOE Course 1 Design of Engineering Experiments The 2 k-p Fractional Factorial Design Motivation for fractional factorials

More information

STA 260: Statistics and Probability II

STA 260: Statistics and Probability II Al Nosedal. University of Toronto. Winter 2017 1 Chapter 7. Sampling Distributions and the Central Limit Theorem If you can t explain it simply, you don t understand it well enough Albert Einstein. Theorem

More information

Institutionen för matematik och matematisk statistik Umeå universitet November 7, Inlämningsuppgift 3. Mariam Shirdel

Institutionen för matematik och matematisk statistik Umeå universitet November 7, Inlämningsuppgift 3. Mariam Shirdel Institutionen för matematik och matematisk statistik Umeå universitet November 7, 2011 Inlämningsuppgift 3 Mariam Shirdel (mash0007@student.umu.se) Kvalitetsteknik och försöksplanering, 7.5 hp 1 Uppgift

More information

CHAPTER 6 A STUDY ON DISC BRAKE SQUEAL USING DESIGN OF EXPERIMENTS

CHAPTER 6 A STUDY ON DISC BRAKE SQUEAL USING DESIGN OF EXPERIMENTS 134 CHAPTER 6 A STUDY ON DISC BRAKE SQUEAL USING DESIGN OF EXPERIMENTS 6.1 INTRODUCTION In spite of the large amount of research work that has been carried out to solve the squeal problem during the last

More information

1. Introduction. Pharmazie 71 (2016) 683 ORIGINAL ARTICLES

1. Introduction. Pharmazie 71 (2016) 683 ORIGINAL ARTICLES Pharmaceutical Chemistry Department 1, The Center for Drug Research and Development (CDRD) 2, Faculty of Pharmacy, British University in Egypt, El-Sherouk City; Analytical Chemistry Department 3, Faculty

More information

FACTOR SCREENING AND RESPONSE SURFACE EXPLORATION

FACTOR SCREENING AND RESPONSE SURFACE EXPLORATION Statistica Sinica 11(2001), 553-604 FACTOR SCREENING AND RESPONSE SURFACE EXPLORATION Shao-Wei Cheng and C. F. J. Wu Academia Sinica and University of Michigan Abstract: Standard practice in response surface

More information

CSCI 688 Homework 6. Megan Rose Bryant Department of Mathematics William and Mary

CSCI 688 Homework 6. Megan Rose Bryant Department of Mathematics William and Mary CSCI 688 Homework 6 Megan Rose Bryant Department of Mathematics William and Mary November 12, 2014 7.1 Consider the experiment described in Problem 6.1. Analyze this experiment assuming that each replicate

More information

THE ROYAL STATISTICAL SOCIETY 2015 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 4

THE ROYAL STATISTICAL SOCIETY 2015 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 4 THE ROYAL STATISTICAL SOCIETY 2015 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 4 The Society is providing these solutions to assist candidates preparing for the examinations in 2017. The solutions are

More information

Chapter 11: Factorial Designs

Chapter 11: Factorial Designs Chapter : Factorial Designs. Two factor factorial designs ( levels factors ) This situation is similar to the randomized block design from the previous chapter. However, in addition to the effects within

More information

Power Order with Differing Numbers of Symbols. This paper represents a contribution to the statistical

Power Order with Differing Numbers of Symbols. This paper represents a contribution to the statistical 8V -558'-f-1 Complete Sets of Orthogonal F-Squares of Prime Power Order with Differing Numbers of Symbols by John P. Mandeli Virginia Commonwealth University Walter T. Federer Cornell University This paper

More information

Multilevel Logic Synthesis Algebraic Methods

Multilevel Logic Synthesis Algebraic Methods Multilevel Logic Synthesis Algebraic Methods Logic Circuits Design Seminars WS2010/2011, Lecture 6 Ing. Petr Fišer, Ph.D. Department of Digital Design Faculty of Information Technology Czech Technical

More information

Fractional Factorial Designs

Fractional Factorial Designs Fractional Factorial Designs ST 516 Each replicate of a 2 k design requires 2 k runs. E.g. 64 runs for k = 6, or 1024 runs for k = 10. When this is infeasible, we use a fraction of the runs. As a result,

More information

Unit 9: Confounding and Fractional Factorial Designs

Unit 9: Confounding and Fractional Factorial Designs Unit 9: Confounding and Fractional Factorial Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand what it means for a treatment to be confounded with blocks Know

More information

Optimal Minimax Controller for Plants with Four Oscillatory Modes Using Gröbner Basis

Optimal Minimax Controller for Plants with Four Oscillatory Modes Using Gröbner Basis 52 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.7, NO.1 February 2009 Optimal Minimax Controller for Plants with Four Oscillatory Modes Using Gröbner Basis Chalie Charoenlarpnopparut

More information

by Christopher Bingham

by Christopher Bingham a D D E!"! #%$&')( $* '$&$,+.-&$0/!12"!"!2 4657 85&$9 ;:=

More information

CS 147: Computer Systems Performance Analysis

CS 147: Computer Systems Performance Analysis CS 147: Computer Systems Performance Analysis Fractional Factorial Designs CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview Overview Overview Example Preparing

More information

A General Criterion for Factorial Designs Under Model Uncertainty

A General Criterion for Factorial Designs Under Model Uncertainty A General Criterion for Factorial Designs Under Model Uncertainty Steven Gilmour Queen Mary University of London http://www.maths.qmul.ac.uk/ sgg and Pi-Wen Tsai National Taiwan Normal University Fall

More information

COM111 Introduction to Computer Engineering (Fall ) NOTES 6 -- page 1 of 12

COM111 Introduction to Computer Engineering (Fall ) NOTES 6 -- page 1 of 12 COM111 Introduction to Computer Engineering (Fall 2006-2007) NOTES 6 -- page 1 of 12 Karnaugh Maps In this lecture, we will discuss Karnaugh maps (K-maps) more formally than last time and discuss a more

More information

Confounding and fractional replication in 2 n factorial systems

Confounding and fractional replication in 2 n factorial systems Chapter 20 Confounding and fractional replication in 2 n factorial systems Confounding is a method of designing a factorial experiment that allows incomplete blocks, i.e., blocks of smaller size than the

More information

19. Blocking & confounding

19. Blocking & confounding 146 19. Blocking & confounding Importance of blocking to control nuisance factors - day of week, batch of raw material, etc. Complete Blocks. This is the easy case. Suppose we run a 2 2 factorial experiment,

More information

Module III Product Quality Improvement. Lecture 4 What is robust design?

Module III Product Quality Improvement. Lecture 4 What is robust design? Module III Product Quality Improvement Lecture 4 What is robust design? Dr. Genichi Taguchi, a mechanical engineer, who has won four times Deming Awards, introduced the loss function concept, which combines

More information

Design of experiment ERT k-p fractional factorial. Miss Hanna Ilyani Zulhaimi

Design of experiment ERT k-p fractional factorial. Miss Hanna Ilyani Zulhaimi + Design of experiment ERT 427 2 k-p fractional factorial Miss Hanna Ilyani Zulhaimi + OUTLINE n Limitation of full factorial design n The concept of fractional factorial, 2 k-p n One-half fraction factorial

More information

Unit 5: Fractional Factorial Experiments at Two Levels

Unit 5: Fractional Factorial Experiments at Two Levels Unit 5: Fractional Factorial Experiments at Two Levels Source : Chapter 4 (sections 4.1-4.3, 4.4.1, 4.4.3, 4.5, part of 4.6). Effect aliasing, resolution, minimum aberration criteria. Analysis. Techniques

More information

Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of

Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of Probability Sampling Procedures Collection of Data Measures

More information

Chapter 5 EXPERIMENTAL DESIGN AND ANALYSIS

Chapter 5 EXPERIMENTAL DESIGN AND ANALYSIS Chapter 5 EXPERIMENTAL DESIGN AND ANALYSIS This chapter contains description of the Taguchi experimental design and analysis procedure with an introduction to Taguchi OA experimentation and the data analysis

More information

Optimal Selection of Blocked Two-Level. Fractional Factorial Designs

Optimal Selection of Blocked Two-Level. Fractional Factorial Designs Applied Mathematical Sciences, Vol. 1, 2007, no. 22, 1069-1082 Optimal Selection of Blocked Two-Level Fractional Factorial Designs Weiming Ke Department of Mathematics and Statistics South Dakota State

More information

Use of DOE methodology for Investigating Conditions that Influence the Tension in Marine Risers for FPSO Ships

Use of DOE methodology for Investigating Conditions that Influence the Tension in Marine Risers for FPSO Ships 1 st International Structural Specialty Conference 1ère Conférence internationale sur le spécialisée sur le génie des structures Calgary, Alberta, Canada May 23-26, 2006 / 23-26 Mai 2006 Use of DOE methodology

More information

Reference: Chapter 6 of Montgomery(8e) Maghsoodloo

Reference: Chapter 6 of Montgomery(8e) Maghsoodloo Reference: Chapter 6 of Montgomery(8e) Maghsoodloo 51 DOE (or DOX) FOR BASE BALANCED FACTORIALS The notation k is used to denote a factorial experiment involving k factors (A, B, C, D,..., K) each at levels.

More information

Great South Channel Habitat Management Area Analysis. Committee tasking

Great South Channel Habitat Management Area Analysis. Committee tasking Great South Channel Habitat Management Area Analysis NEFMC Habitat Committee Meeting March 19, 2013 Salem, MA Committee tasking MOTION 5, (McKenzie, Alexander) from 12 4 12 meeting Move that the Committee

More information

Statistical Design and Analysis of Experiments Part Two

Statistical Design and Analysis of Experiments Part Two 0.1 Statistical Design and Analysis of Experiments Part Two Lecture notes Fall semester 2007 Henrik Spliid nformatics and Mathematical Modelling Technical University of Denmark List of contents, cont.

More information

Unit 6: Fractional Factorial Experiments at Three Levels

Unit 6: Fractional Factorial Experiments at Three Levels Unit 6: Fractional Factorial Experiments at Three Levels Larger-the-better and smaller-the-better problems. Basic concepts for 3 k full factorial designs. Analysis of 3 k designs using orthogonal components

More information

Two-Level Fractional Factorial Design

Two-Level Fractional Factorial Design Two-Level Fractional Factorial Design Reference DeVor, Statistical Quality Design and Control, Ch. 19, 0 1 Andy Guo Types of Experimental Design Parallel-type approach Sequential-type approach One-factor

More information

20g g g Analyze the residuals from this experiment and comment on the model adequacy.

20g g g Analyze the residuals from this experiment and comment on the model adequacy. 3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. One-way ANOVA Source DF SS MS F P Factor 3 36.15??? Error??? Total 19 196.04 3.11. A pharmaceutical

More information

Great South Channel Habitat Management Area Analysis

Great South Channel Habitat Management Area Analysis Great South Channel Habitat Management Area Analysis NEFMC Habitat Committee Meeting March 19, 2013 Salem, MA Note that this is the version presented at the meeting modified from version previously posted

More information

7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology)

7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology) 7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Introduction Response surface methodology,

More information

EXISTENCE AND CONSTRUCTION OF RANDOMIZATION DEFINING CONTRAST SUBSPACES FOR REGULAR FACTORIAL DESIGNS

EXISTENCE AND CONSTRUCTION OF RANDOMIZATION DEFINING CONTRAST SUBSPACES FOR REGULAR FACTORIAL DESIGNS Submitted to the Annals of Statistics EXISTENCE AND CONSTRUCTION OF RANDOMIZATION DEFINING CONTRAST SUBSPACES FOR REGULAR FACTORIAL DESIGNS By Pritam Ranjan, Derek R. Bingham and Angela M. Dean, Acadia

More information

LECTURE 10: LINEAR MODEL SELECTION PT. 1. October 16, 2017 SDS 293: Machine Learning

LECTURE 10: LINEAR MODEL SELECTION PT. 1. October 16, 2017 SDS 293: Machine Learning LECTURE 10: LINEAR MODEL SELECTION PT. 1 October 16, 2017 SDS 293: Machine Learning Outline Model selection: alternatives to least-squares Subset selection - Best subset - Stepwise selection (forward and

More information

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Response Surface Methodology IV

Response Surface Methodology IV LECTURE 8 Response Surface Methodology IV 1. Bias and Variance If y x is the response of the system at the point x, or in short hand, y x = f (x), then we can write η x = E(y x ). This is the true, and

More information

Lec 10: Fractions of 2 k Factorial Design

Lec 10: Fractions of 2 k Factorial Design December 5, 2011 Fraction of 2 k experiments Screening: Some of the factors may influence the results. We want to figure out which. The number of combinations, 2 k, is too large for a complete investigation.

More information

Robust Design: An introduction to Taguchi Methods

Robust Design: An introduction to Taguchi Methods Robust Design: An introduction to Taguchi Methods The theoretical foundations of Taguchi Methods were laid out by Genichi Taguchi, a Japanese engineer who began working for the telecommunications company,

More information

The 2 k Factorial Design. Dr. Mohammad Abuhaiba 1

The 2 k Factorial Design. Dr. Mohammad Abuhaiba 1 The 2 k Factorial Design Dr. Mohammad Abuhaiba 1 HoweWork Assignment Due Tuesday 1/6/2010 6.1, 6.2, 6.17, 6.18, 6.19 Dr. Mohammad Abuhaiba 2 Design of Engineering Experiments The 2 k Factorial Design Special

More information

CS 5014: Research Methods in Computer Science

CS 5014: Research Methods in Computer Science Computer Science Clifford A. Shaffer Department of Computer Science Virginia Tech Blacksburg, Virginia Fall 2010 Copyright c 2010 by Clifford A. Shaffer Computer Science Fall 2010 1 / 254 Experimental

More information

Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks.

Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks. 58 2. 2 factorials in 2 blocks Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks. Some more algebra: If two effects are confounded with

More information

USE OF COMPUTER EXPERIMENTS TO STUDY THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS

USE OF COMPUTER EXPERIMENTS TO STUDY THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS USE OF COMPUTER EXPERIMENTS TO STUDY THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS Seshadev Padhi, Manish Trivedi and Soubhik Chakraborty* Department of Applied Mathematics

More information

SUM-OF-SQUARES ORTHOGONALITY AND COMPLETE SETS

SUM-OF-SQUARES ORTHOGONALITY AND COMPLETE SETS 3«- )h37-frl 1 SUM-OF-SQUARES ORTHOGONALITY AND COMPLETE SETS by Walter T. Federer ABSTRACT The concept of sum-of-squares orthogonality is explained in terms of what is known about factorial treatment

More information

Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur

Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Design and Analysis of Experiments Prof. Jhareshwar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Lecture 51 Plackett Burman Designs Hello, welcome. We

More information

choosedef2(7,4,all:t) K

choosedef2(7,4,all:t) K i!"! a ` a c a ``` `aaa ``` aaa ``` `!ccc j'$k$ 1 C l ; B-?hm 4noqsr $h t=;2 4nXu ED4+* J D98 B v-,/. = $-r

More information

Design and Analysis of

Design and Analysis of Design and Analysis of Multi-Factored Experiments Module Engineering 7928-2 Two-level Factorial Designs L. M. Lye DOE Course 1 The 2 k Factorial Design Special case of the general factorial design; k factors,

More information

SIX SIGMA IMPROVE

SIX SIGMA IMPROVE SIX SIGMA IMPROVE 1. For a simplex-lattice design the following formula or equation determines: A. The canonical formula for linear coefficients B. The portion of each polynomial in the experimental model

More information

PURPLE COMET MATH MEET April 2012 MIDDLE SCHOOL - SOLUTIONS

PURPLE COMET MATH MEET April 2012 MIDDLE SCHOOL - SOLUTIONS PURPLE COMET MATH MEET April 2012 MIDDLE SCHOOL - SOLUTIONS Copyright c Titu Andreescu and Jonathan Kane Problem 1 Evaluate 5 4 4 3 3 2 2 1 1 0. Answer: 549 The expression equals 625 64 9 2 1 = 549. Problem

More information

Topics in Experimental Design

Topics in Experimental Design Ronald Christensen Professor of Statistics Department of Mathematics and Statistics University of New Mexico Copyright c 2016 Topics in Experimental Design Springer Preface An extremely useful concept

More information

Some Nonregular Designs From the Nordstrom and Robinson Code and Their Statistical Properties

Some Nonregular Designs From the Nordstrom and Robinson Code and Their Statistical Properties Some Nonregular Designs From the Nordstrom and Robinson Code and Their Statistical Properties HONGQUAN XU Department of Statistics, University of California, Los Angeles, CA 90095-1554, U.S.A. (hqxu@stat.ucla.edu)

More information

Moment Aberration Projection for Nonregular Fractional Factorial Designs

Moment Aberration Projection for Nonregular Fractional Factorial Designs Moment Aberration Projection for Nonregular Fractional Factorial Designs Hongquan Xu Department of Statistics University of California Los Angeles, CA 90095-1554 (hqxu@stat.ucla.edu) Lih-Yuan Deng Department

More information

A Fast Algorithm for Protein Structural Comparison

A Fast Algorithm for Protein Structural Comparison A Fast Algorithm for Protein Structural Comparison Sheng-Lung Peng and Yu-Wei Tsay Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien 974, Taiwan Abstract

More information

Response Surface Methodology

Response Surface Methodology Response Surface Methodology Process and Product Optimization Using Designed Experiments Second Edition RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona

More information

ESTIMATION METHODS FOR MISSING DATA IN UN-REPLICATED 2 FACTORIAL AND 2 FRACTIONAL FACTORIAL DESIGNS

ESTIMATION METHODS FOR MISSING DATA IN UN-REPLICATED 2 FACTORIAL AND 2 FRACTIONAL FACTORIAL DESIGNS Journal of Statistics: Advances in Theory and Applications Volume 5, Number 2, 2011, Pages 131-147 ESTIMATION METHODS FOR MISSING DATA IN k k p UN-REPLICATED 2 FACTORIAL AND 2 FRACTIONAL FACTORIAL DESIGNS

More information

RESPONSE SURFACE MODELLING, RSM

RESPONSE SURFACE MODELLING, RSM CHEM-E3205 BIOPROCESS OPTIMIZATION AND SIMULATION LECTURE 3 RESPONSE SURFACE MODELLING, RSM Tool for process optimization HISTORY Statistical experimental design pioneering work R.A. Fisher in 1925: Statistical

More information

A Statistical Approach to the Study of Qualitative Behavior of Solutions of Second Order Neutral Differential Equations

A Statistical Approach to the Study of Qualitative Behavior of Solutions of Second Order Neutral Differential Equations Australian Journal of Basic and Applied Sciences, (4): 84-833, 007 ISSN 99-878 A Statistical Approach to the Study of Qualitative Behavior of Solutions of Second Order Neutral Differential Equations Seshadev

More information

Process Robustness Studies

Process Robustness Studies Process Robustness Studies ST 435/535 Background When factors interact, the level of one can sometimes be chosen so that another has no effect on the response. If the second factor is controllable in a

More information

APPLICATION OF DISCRETE DISTRIBUTIONS IN QUALITY CONTROL A THESIS. Presented to. The Faculty of the Division of Graduate. Studies and Research

APPLICATION OF DISCRETE DISTRIBUTIONS IN QUALITY CONTROL A THESIS. Presented to. The Faculty of the Division of Graduate. Studies and Research APPLICATION OF DISCRETE DISTRIBUTIONS IN QUALITY CONTROL A THESIS Presented to The Faculty of the Division of Graduate Studies and Research By Milton Richard Scheffler In Partial Fulfillment of the Requirements

More information

Optimization of Muffler and Silencer

Optimization of Muffler and Silencer Chapter 5 Optimization of Muffler and Silencer In the earlier chapter though various numerical methods are presented, they are not meant to optimize the performance of muffler/silencer for space constraint

More information

Response Surface Methodology:

Response Surface Methodology: Response Surface Methodology: Process and Product Optimization Using Designed Experiments RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona State University

More information