Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 1.

Slides:



Advertisements
Similar presentations
Design of Experiments Lecture I
Advertisements

Sampling plans for linear regression
Introduction to Design of Experiments by Dr Brad Morantz
Process Control: Designing Process and Control Systems for Dynamic Performance Chapter 6. Empirical Model Identification Copyright © Thomas Marlin 2013.
IE341 Problems. 1.Nuisance effects can be known or unknown. (a) If they are known, what are the ways you can deal with them? (b) What happens if they.
Experimental Design, Response Surface Analysis, and Optimization
Sampling plans Given a domain, we can reduce the prediction error by good choice of the sampling points The choice of sampling locations is called “design.
Optimisation.
Rethinking Steepest Ascent for Multiple Response Applications Robert W. Mee Jihua Xiao University of Tennessee.
Empirical Maximum Likelihood and Stochastic Process Lecture VIII.
1 Chapter 5: Custom Designs 5.1 Custom Design Generation 5.2 Custom Response Surface Designs.
11.1 Introduction to Response Surface Methodology
Instructor: Mircea Nicolescu Lecture 13 CS 485 / 685 Computer Vision.
Response Surface Method Principle Component Analysis
Design and Analysis of Experiments
14-1 Introduction An experiment is a test or series of tests. The design of an experiment plays a major role in the eventual solution of the problem.
Industrial Applications of Response Surface Methodolgy John Borkowski Montana State University Pattaya Conference on Statistics Pattaya, Thailand.
/ department of mathematics and computer science DS01 Statistics 2 for Chemical Engineering lecture 3
Development of Empirical Models From Process Data
Lecture 17 Today: Start Chapter 9 Next day: More of Chapter 9.
Lecture 19 Today: More Chapter 9 Next day: Finish Chapter 9 Assignment: Chapter 9: 17.
Statistics 2 for Chemical Engineering lecture 4
Design-Expert version 71 What’s New in Design-Expert version 7 Factorial and RSM Design Pat Whitcomb November, 2006.
Design and Analysis of Engineering Experiments
10.1 Chapter 10 Optimization Designs Optimization Designs CS RO R Focus: A Few Continuous Factors Output: Best Settings Reference: Box, Hunter &
Chapter 1Based on Design & Analysis of Experiments 7E 2009 Montgomery 1 Design and Analysis of Engineering Experiments Ali Ahmad, PhD.
1 14 Design of Experiments with Several Factors 14-1 Introduction 14-2 Factorial Experiments 14-3 Two-Factor Factorial Experiments Statistical analysis.
Design and Analysis of Multi-Factored Experiments
VIII. Introduction to Response Surface Methodology
RESPONSE SURFACE METHODOLOGY (R S M)
Chapter 8Design and Analysis of Experiments 8E 2012 Montgomery 1 Design of Engineering Experiments The 2 k-p Fractional Factorial Design Text reference,
Chapter 4: Response Surface Methodology
ITK-226 Statistika & Rancangan Percobaan Dicky Dermawan
DOX 6E Montgomery1 Design of Engineering Experiments Part 7 – The 2 k-p Fractional Factorial Design Text reference, Chapter 8 Motivation for fractional.
Non-Linear Models. Non-Linear Growth models many models cannot be transformed into a linear model The Mechanistic Growth Model Equation: or (ignoring.
Industrial Applications of Experimental Design John Borkowski Montana State University University of Economics and Finance HCMC, Vietnam.
Statistical Design of Experiments
Nonlinear programming Unconstrained optimization techniques.
Non-Linear Models. Non-Linear Growth models many models cannot be transformed into a linear model The Mechanistic Growth Model Equation: or (ignoring.
Centerpoint Designs Include n c center points (0,…,0) in a factorial design Include n c center points (0,…,0) in a factorial design –Obtains estimate of.
Design of Engineering Experiments Part 5 – The 2k Factorial Design
Engineering Statistics ENGR 592 Prepared by: Mariam El-Maghraby Date: 26/05/04 Design of Experiments Plackett-Burman Box-Behnken.
Geographic Information Science
Chapter 6Design & Analysis of Experiments 8E 2012 Montgomery 1 Design of Engineering Experiments Part 5 – The 2 k Factorial Design Text reference, Chapter.
Design of Experiments DoE Antonio Núñez, ULPGC. Objectives of DoE in D&M Processes, Process Investigation, Product and Process Q-improvement, Statistical.
SAMO 2007, 21 Jun 2007 Davood Shahsavani and Anders Grimvall Linköping University.
Chapter 6Design & Analysis of Experiments 7E 2009 Montgomery 1 Design of Engineering Experiments Two-Level Factorial Designs Text reference, Chapter 6.
International Conference on Design of Experiments and Its Applications July 9-13, 2006, Tianjin, P.R. China Sung Hyun Park, Hyuk Joo Kim and Jae-Il.
DOX 6E Montgomery1 Design of Engineering Experiments Part 8 – Overview of Response Surface Methods Text reference, Chapter 11, Sections 11-1 through 11-4.
The American University in Cairo Interdisciplinary Engineering Program ENGR 592: Probability & Statistics 2 k Factorial & Central Composite Designs Presented.
CHAPTER 17 O PTIMAL D ESIGN FOR E XPERIMENTAL I NPUTS Organization of chapter in ISSO –Background Motivation Finite sample and asymptotic (continuous)
Additional Topics in Prediction Methodology. Introduction Predictive distribution for random variable Y 0 is meant to capture all the information about.
L. M. LyeDOE Course1 Design and Analysis of Multi-Factored Experiments Response Surface Methodology.
Non-Linear Models. Non-Linear Growth models many models cannot be transformed into a linear model The Mechanistic Growth Model Equation: or (ignoring.
Lecture 18 Today: More Chapter 9 Next day: Finish Chapter 9.
Performance Surfaces.
Design and Analysis of Experiments (7) Response Surface Methods and Designs (2) Kyung-Ho Park.
1 Design of Engineering Experiments – The 2 k Factorial Design Text reference, Chapter 6 Special case of the general factorial design; k factors, all at.
Introduction to Statistical Quality Control, 4th Edition Chapter 13 Process Optimization with Designed Experiments.
Computacion Inteligente Least-Square Methods for System Identification.
11 Designs for the First Order Model “Orthogonal first order designs” minimize the variance of If the off diagonal elements of X´X are all zero then the.
Axially Variable Strength Control Rods for The Power Maneuvering of PWRs KIM, UNG-SOO Dept. of Nuclear and Quantum Engineering.
A Primer on Running Deterministic Experiments
Design of Engineering Experiments Part 5 – The 2k Factorial Design
Chapter 14 Introduction to Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2012  John Wiley & Sons, Inc.
ENM 310 Design of Experiments and Regression Analysis Chapter 3
What are optimization methods?
Centerpoint Designs Include nc center points (0,…,0) in a factorial design Obtains estimate of pure error (at center of region of interest) Tests of curvature.
14 Design of Experiments with Several Factors CHAPTER OUTLINE
Uncertainty Propagation
Presentation transcript:

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 1

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 2 Text reference, Chapter 11 Primary focus of previous chapters is factor screening –Two-level factorials, fractional factorials are widely used Objective of RSM is optimization RSM dates from the 1950s; early applications in chemical industry Modern applications of RSM span many industrial and business settings

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 3 Response Surface Methodology Collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables Objective is to optimize the response

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 4 Steps in RSM 1.Find a suitable approximation for y = f(x) using LS {maybe a low – order polynomial} 2.Move towards the region of the optimum 3.When curvature is found find a new approximation for y = f(x) {generally a higher order polynomial} and perform the “Response Surface Analysis”

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 5 Response Surface Models Screening Steepest ascent Optimization

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 6 RSM is a Sequential Procedure Factor screening Finding the region of the optimum Modeling & Optimization of the response

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 7 The Method of Steepest Ascent Text, Section 11.2 A procedure for moving sequentially from an initial “guess” towards to region of the optimum Based on the fitted first- order model Steepest ascent is a gradient procedure

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 8 Example 11.1: An Example of Steepest Ascent

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 9

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 10

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 11

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 12

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 13

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 14

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 15

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 16 Points on the path of steepest ascent are proportional to the magnitudes of the model regression coefficients The direction depends on the sign of the regression coefficient Step-by-step procedure:

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 17 Second-Order Models in RSM These models are used widely in practice The Taylor series analogy Fitting the model is easy, some nice designs are available Optimization is easy There is a lot of empirical evidence that they work very well

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 18

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 19

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 20

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 21 Characterization of the Response Surface Find out where our stationary point is Find what type of surface we have –Graphical Analysis –Canonical Analysis Determine the sensitivity of the response variable to the optimum value –Canonical Analysis

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 22 Finding the Stationary Point After fitting a second order model take the partial derivatives with respect to the x i ’s and set to zero –δy / δx 1 =... = δy / δx k = 0 Stationary point represents… –Maximum Point –Minimum Point –Saddle Point

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 23 Stationary Point

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 24 Canonical Analysis Used for sensitivity analysis and stationary point identification Based on the analysis of a transformed model called: canonical form of the model Canonical Model form: y = y s + λ 1 w λ 2 w λ k w k 2

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 25

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 26 Eigenvalues The nature of the response can be determined by the signs and magnitudes of the eigenvalues –{e} all positive: a minimum is found –{e} all negative: a maximum is found –{e} mixed: a saddle point is found Eigenvalues can be used to determine the sensitivity of the response with respect to the design factors The response surface is steepest in the direction (canonical) corresponding to the largest absolute eigenvalue

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 27

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 28

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 29

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 30

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 31

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 32

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 33 Ridge Systems

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 34

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 35

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 36

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 37

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 38 Overlay Contour Plots

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 39 Mathematical Programming Formulation

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 40 Desirability Function Method

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 41

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 42

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 43

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 44

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 45 Addition of center points is usually a good idea

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 46

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 47 The Rotatable CCD

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 48

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 49 The Box-Behnken Design

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 50 A Design on A Cube – The Face-Centered CCD

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 51 Note that the design isn’t rotatable but the prediction variance is very good in the center of the region of experimentation

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 52 Other Designs Equiradial designs (k = 2 only) The small composite design (SCD) –Not a great choice because of poor prediction variance properties Hybrid designs –Excellent prediction variance properties –Unusual factor levels Computer-generated designs

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 53

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 54

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 55

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 56 Blocking in a Second-Order Design

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 57

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 58 Computer-Generated (Optimal) Designs These designs are good choices whenever –The experimental region is irregular –The model isn’t a standard one –There are unusual sample size or blocking requirements These designs are constructed using a computer algorithm and a specified “optimality criterion” Many “standard” designs are either optimal or very nearly optimal

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 59

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 60

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 61

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 62 Which Criterion Should I Use? For fitting a first-order model, D is a good choice –Focus on estimating parameters –Useful in screening For fitting a second-order model, I is a good choice –Focus on response prediction –Appropriate for optimization

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 63 Algorithms Point exchange –Requires a candidate set of points –The design is chosen from the candidate set –Random start, several (many) restarts to ensure that a highly efficient design is found Coordinate exchange –No candidate set required –Search over each coordinate one-at-a-time –Many random starts used

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 64 The Adhesive Pull-Off Force Experiment – a “Standard” Design

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 65 A D-Optimal Design

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 66 Relative efficiency of the standard inscribed design The standard design would have to be replicated approximately twice to estimate the parameters as precisely as the optimal design

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 67

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 68

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 69

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 70

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 71

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 72

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 73

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 74

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 75

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 76

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 77

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 78 Designs for Computer Experiments Optimal designs are appropriate if a polynomial model is used Space-filling designs are also widely used for non-polynomial models –Latin hypercube designs –Sphere-packing designs –Uniform designs –Maximum entropy designs

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 79

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 80

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 81 The Gaussian Process Model Spatial correlation model Interpolates the data One parameter for each factor – more parsimonious that polynomials Often a good choice for a deterministic computer model

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 82

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 83

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 84

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 85

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 86

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 87

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 88

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 89

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 90 Mixture Models

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 91

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 92

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 93

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 94 Constraints

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 95

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 96

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 97

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 98

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 99

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 100

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 101

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 102

Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 103 Evolutionary Operation (EVOP) An experimental deign based technique for continuous monitoring and improvement of a process Small changes are continuously introduced in the important variables of a process and the effects evaluated The 2-level factorial is recommended There are usually only 2 or 3 factors considered EVOP has not been widely used in practice The text has a complete example