Appendix IV Experimental Design

Size: px
Start display at page:

Download "Appendix IV Experimental Design"

Transcription

1 Experimental Design The aim of pharmaceutical formulation and development is to develop an acceptable pharmaceutical formulation in the shortest possible time, using minimum number of working hours and raw materials. The formula developed by the formulation and development center is first tried at the pilot scale and then manufacture scale. Only minor changes are to be made during scale-up. Thus, it is very ideal to study the formulation from all perspectives at laboratory levels. In addition to the art of formulation, a statistical technique is available that can aid in the pharmacist s choice of formulation components, which can optimize one or more formulation attributes. A very efficient way to enhance the value of research and to minimize the process development time is through design experiment. The need to develop this design because traditional experiments involve a good deal of efforts and time, especially where complex formulations are to be developed. The statistical problem solving approach uses a series of small carefully designed experiments. We sometimes call the statistical approach strategic experimentation or iterative problem solving strategy. We also call this the stop look and listen approach to experimentation. Analyze the results of few experiments and then plan the next experiments. Any statistical design consist of three small and efficient experiments, namely a screening experiments where from many factors affecting the process few important factors are identified, then an optimization experiment where a predictive model is build for the few factors in the region of optimum and finally a verification experiment where the results is confirmed at the predicted setting. In the present work factorial design was used for the development of effective, functional and perfect dosage form. The help of systematic formulation approach is taken to get detailed knowledge on the formulation. 275

2 In the present study, 23 and factorial design and response surface methodology like central composite designs and Box-Behnken design were used. Hence, only these designs are discussed in details. IV-A. FACTORIAL DESIGNS [1-3] Factorial designs are used in experiments when the effects of different factors or conditions, on experiment results are to be elucidated. Factorial designs are the design of choice of simultaneous determination of the effects of several factors and their interaction. Factors may be qualitative or quantitative. The levels of an each factor are the values or designations assigned to combinations, of all levels, of all factors. The effects of a factor are the change in response caused by varying the levels(s) of the factor. The important objective of a factorial experiment is to characterize the effect of changing the levels of the factor or combination of factors on the response variable. Predictions based on results of an undersigned experiment will be more variable than those, which could be obtained in a designed experiment, in particular factorial design. The optimization procedure is facilitated by construction of an equation that describes the experimental results as a function of the factor levels. A polynomial equation can be constructed, where the coefficients in the equation are related to the effects and interaction of the factors. The goal of pharmaceutical formulation and development is to develop acceptable pharmaceutical formulation in the shortest possible time using minimum time and raw materials Optimization by experimental design leads to the evolution of a statistically valid model to understand the relationship between independent and dependent variables. The equation constructed form 2n factorial experiment is in the following from. Y= B0+B,Xi+B2X2+ B3X3+B12X1X2+B13X1X3+B23X2X3+B123X1X2X3 Where, Y= the measured response Xj= level of i,h factor (independent variable) 276

3 Appendix I V Bj = the regression coefficient for the ith independent variable. B0 = intercept The magnitudes of the coefficients represent the relative importance of each factor. Once the polynomial equation has been established, an optimum formulation can be found out by grid analysis. With the use of computer a grid method can be used to identify optimum regions, and response surfaces may be depicted. A computer can calculate the response based on equation at many combinations of factor levels. The formulation whose response has optimal characteristics based on the experimenter s specification is then chosen. Advantages of Factorial Design: In absence of interaction, they have maximum efficiency in estimating main effects. Maximum use is made of the data, since all main effects and interaction are calculated from the data Since factors effects are measured over varying levels of other factors, conclusions apply to wide range of condition. Factorial designs are orthogonal; all estimated effect and interaction are independent of effect of other factors. If interaction occur; factorial designs are necessary to reveal and identity the interaction. More information is obtained with less work. The most important attribute of this design is that the effects are measured with maximum precision. 277

4 Applications of Factorial Design: It helps and interprets the mechanism of an experimental system. It is very useful in an industrial manufacturing operation because it recommend or implement, a practical procedure or a set of condition It provides guidance for further experiment It also useful for the drug-excipinents compatibility study. In most situations, where one is interested in the effect of various factors or condition on some experimental outcome, factorial designs will be optimal. Factorial design used is either full or fractional. Full factorial design is a design in which every setting of every factor appears with every setting of every other factor is called as a full factorial design. When experiments are with a large number of factors and /or a large number of levels for the factors, the number of factors needed to complete factorial design is also large. Thus, application of full factorial design becomes costly and time consuming. In this case, fractional factorial design is utilized for optimization. In fractional factorial designs, the negligible and important factors are indistinguishable, and thus, confounded. Full Factorial Design: 23 factorial designs: A common experimental design is one with all input factors set at two levels each. These levels are called 'high' and 'low' or '+!' and '-1', respectively. A design with all possible high/low combinations of all the input factors is called a full factorial design in two levels. If there are k factors, each at 2 levels, a full factorial design has 2k runs. 278

5 Table 1: Runs in 2k full Factorial Design Number of Factors Number of Runs In 23 full factorial designs three factors and two levels are used to achieve the proper result. This implies eight runs (not counting replications or center point runs). Graphically, the 2 design is represented by the cube shown in figure. The arrows show the direction of increase of the factors. The numbers 1 through 8 at the corners of the design box reference the Standard Order of runs. 6 8 Figure 1:23 two-level, full factorial design; factors Xi, X2, X3 279

6 4 The design of 2 factorial design is given in Table 2. Table 2: Design of 23' factorial design Trial X, x2 x l l i l *1 -l l +i i +i +1 The design of 32 factorial design is as given in Table 3. In 32 full factorial designs two factors and three levels are used. Total 9 trials are made if this design is employed. Table 3: 32 Factorial Design Trial X, x

7 IV-B. RESPONSE SURFACE METHODOLOGY These designs can be classified as follows: Second order designs for spherical domain Second order designs for the cubic domain Central composite designs 3k factorial designs Special designs for cubic domain Central Composite Designs [4-6] Central composite designs are response surface designs that can fit a full quadratic model. One central composite design consists of cube points at the comers of a unit cube that is the product of the intervals [-1, 1], stars points along the axes at or outside the cube, and centre points at the origin. The design has three basic parts: The design consists of three distinct sets of experimental runs: 1. A factorial (perhaps fractional) design in the factors studied, each having two levels(2k F); 2. A set of centre points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often repeated in order to improve the precision of the experiment; thus the term is central composite, and 3. A set of axial points, experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range(i.e. starpoints, 2K).. All factors are varied in this way. 281

8 The number of experimental trials (N) in a composite design is given by: N = 2k f + 2K + C Where, K = no. of variables F = fraction of the full factorial and C = number of center point replicates Design Matrix: The design matrix for a central composite design experiment involving k factors is derived from a matrix, d, containing the following three different parts corresponding to the three types of experimental runs: 1. The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and The matrix C from the centre points, denoted in coded variables as (0,0,0,...,0), where there are k zeros. 3. A matrix E from the axial points, with 2k rows. Each factor is sequentially placed at ±a and all other factors are at zero. The value of a is determined by the designer; while arbitrary, some values may give the design desirable properties. This part would look like: a 0 0 a a 0 E = 0 -a a a 282

9 Central Composite Designs are of three types: Circumscribed (CCC) designs are as described above. Inscribed (CCI) designs are as described above, but scaled so the star points take the values -1 and +1, and the cube points lie in the interior of the cube. Faced (CCF) designs have the star points on the faces of the cube. Faced designs have three levels per factor, in contrast with the other types, which have five levels per factor. It is shown in Figure 2. CCD designs start with a factorial or fractional factorial design (with center points) and add "star" points to estimate curvature Similarly, the number of center point runs the design is also depends on certain properties required for the design. Characteristic of Central Composite Designs 1. Central Composite Circumscribed (CCC): This design is the original form of the central composite design. The star points are at some distance a from the center based on the properties desired for the design and the number of factors in the design. The star points establish new extremes for the low and high settings for all factors. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. CCC designs provide high quality predictions over the entire design space, but require factor settings outside the range of the factors in the factorial part. 2. Central Composite Inscribed (CCI): For those situations in which the limits specified for factor settings are truly limits, the CCI design uses the factor settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by a to generate the CCI design). This design also requires 5 levels of each factor. CCI designs use only points within the factor ranges originally specified, but do not provide the same high quality prediction over the entire space compared to the CCC. Requires 5 levels of each factor. 283

10 3. Central Composite Face Centered (CCF): In this design the star points are at the center of each face of the factorial space, so a = ± 1. This variety requires 3 levels of each factor. Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design. CCF designs provide relatively high quality predictions over the entire design space and do not require using points outside the original factor range. However, they give poor precision for estimating pure quadratic coefficients. These designs require fewer treatment combinations than a central composite design in cases involving 3 or 4 factors. Table 4: Design Matrix for Central Composite Design Runs X, x "hi Box-Behnken Designs 17-12] Screening and optimizing process for dosage form development can be simplified by the use of a statistical design that requires only a small number of experiments, thereby eliminating the need for time-consuming and detailed experimental trials. Response surface methodology (RSM) is one such approach. It is used when only a few significant factors are involved in optimization. Different types of RSM designs include 3-level factorial design, central composite design (CCD), Box-Behnken design, and D-optimal design. The comparison of number of runs required for a given number of factors for various Central Composite design and Box-Behnken design is given in table

11 Table 5: Comparison of numbers of runs for Central Composite and Box-Behnken Designs Number of Box-behnken Central Composite Design Factors Design 2 13 (5 center points) (6 center point runs) (6 center point runs) (fractional factorial) or 52 (full factorial) (fractional factorial) or 91 (full factorial) 54 The runs or trials of the composite design experiments consist of all combination of all levels of all factors. The effect of a factor is the change in response caused by varying the level(s) of the factor chosen. Box-Behnken design requires fewer runs (15 runs) in a 3-factor experimental design A3-factor, 3-level design would require a total of 27 unique runs without any repetitions and a total of 30 runs with 3 repetitions. A 3-factor, 3-level design used is suitable for exploring quadratic response surfaces and constructing second-order polynomial models. It is basically a modified central composite experimental design, which, is an independent, rotatable or nearly rotatable quadratic design (contains no embedded factorial or fractional factorial design),. It consists of 1. Center points, and 2. Points lying on one sphere, equally distant from the center point. The latter points consists of small two-level full factorials where some factors are fixed at their center values. The number of center points is chosen to establish rotatability. The nonlinear quadratic model generated by the design is expressed as follows: The optimization procedure is facilitated by construction of an equation that describes the experimental results as a function of the factor levels. A polynomial equation can be constructed, where the coefficients in the equation are related to the effects and interactions of the factors. Y = b0 + b,x, + b2x2 + b3x3 + b12x,x2- b13x,x3+ b23x2x3+ b,,x,2 + b22 X22 + b33x32 Where Y= Estimated Response associated with each factor level combination. 285

12 Xi, X2 and X3 =Independent variables bi to b33 = Regression coefficients The magnitude of the coefficients represents the relative importance of each factor. Once the polynomial equation has been established, an optimum formulation can be found out by grid analysis. With the use of computers a grid method can be used to identify optimum regions and response surfaces may be depicted. A computer can calculate the response based on equation at many combinations of factor levels. The formulation whose response has optimal characteristics based on the experimenter s specification. Table 6:Design Matrix for 3 factors Box-Behnken design Trials Variable levels in coded form X, x2 X T

13 REFERENCES: 1. Bolton S., Bon C., Pharmaceutical Statistics: Practical and clinical application, 2nd Ed., Marcel Dekker Inc., NY, 1990: ; Franz R.M., Browne J.E., and Lewis A.R.; Experimental design, modeling an optimization strategies for product and process development: In Libermann, H.A. Riger, M.M., Banker, G.S., (Eds.), Pharmaceutical dosage form: Disperse systems (Volume I), Marcel Dekker, NY, 1988: Lewis,GA, Mathieu D, Phan-Tan-Luu R. Pharmaceutical Experimental Design Marcel Dekker Inc., New York, 1999, html, Accessed on 28th August, Lewis GA, Optimization Methods in Encyclopedia of Pharmaceutical technology, Edited by Swarbrick, J., Boylan, J.C., Marcel Dekker Inc., New York, 2nd edition, 2002; Box GEP, Wilson KB. On the experimental attainment of optimum multifactorial conditions. Royal Statistics Society. 1951; 13: Singh SK, Dodge J, Durrani MJ, Khan MA. Optimization and characterization of controlled release pellets coated with experimental latex: I. Anionic drug. Int J Pharm. 1995; 125: Sanchez-Lafuente C, Furlanetto S, Femandez-Arevalo M, et al. Didanosine extendedrelease matrix tablets: optimization of formulation variables using statistical experimental design. Int J Pharm.2002; 237: Ragonese R, Macka M, Hughes J, Petocz P. The use of the Box- Behnken experimental design in the optimization and robustness testing of a capillary electrophoresis method for the analysis of ethambutol hydrochloride in a pharmaceutical formulation. J Pharm Biomed Anal. 2002: 27: Box GEP, Behnken DW. Some new three level designs for the study of quantitative variables. Technometrics. 1960;2: Wilson WI, Peng Y, Augsburger LL. Comparison of statistical analysis and Bayesian networks in the evaluation of dissolution performance of BCS class II model drugs. J. Pharma. Sci., 94(12):

Experimental Design and Optimization

Experimental Design and Optimization . Experimental Design Stages a) Identifying the factors which may affect the results of an experiment; b) Designing the experiment so that the effects of uncontrolled factors are minimized; c) Using statistical

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

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

Analysis of Variance and Design of Experiments-II

Analysis of Variance and Design of Experiments-II Analysis of Variance and Design of Experiments-II MODULE VIII LECTURE - 36 RESPONSE SURFACE DESIGNS Dr. Shalabh Department of Mathematics & Statistics Indian Institute of Technology Kanpur 2 Design for

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

8 RESPONSE SURFACE DESIGNS

8 RESPONSE SURFACE DESIGNS 8 RESPONSE SURFACE DESIGNS Desirable Properties of a Response Surface Design 1. It should generate a satisfactory distribution of information throughout the design region. 2. It should ensure that the

More information

MATH602: APPLIED STATISTICS

MATH602: APPLIED STATISTICS MATH602: APPLIED STATISTICS Dr. Srinivas R. Chakravarthy Department of Science and Mathematics KETTERING UNIVERSITY Flint, MI 48504-4898 Lecture 10 1 FRACTIONAL FACTORIAL DESIGNS Complete factorial designs

More information

DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi

DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi + DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi + Outline n Definition of Response Surface Methodology n Method of Steepest Ascent n Second-Order Response Surface

More information

D-Optimal Designs for Second-Order Response Surface Models with Qualitative Factors

D-Optimal Designs for Second-Order Response Surface Models with Qualitative Factors Journal of Data Science 920), 39-53 D-Optimal Designs for Second-Order Response Surface Models with Qualitative Factors Chuan-Pin Lee and Mong-Na Lo Huang National Sun Yat-sen University Abstract: Central

More information

Practical Statistics for the Analytical Scientist Table of Contents

Practical Statistics for the Analytical Scientist Table of Contents Practical Statistics for the Analytical Scientist Table of Contents Chapter 1 Introduction - Choosing the Correct Statistics 1.1 Introduction 1.2 Choosing the Right Statistical Procedures 1.2.1 Planning

More information

Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition

Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition Brochure More information from http://www.researchandmarkets.com/reports/705963/ Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition Description: Identifying

More information

Chemometrics Unit 4 Response Surface Methodology

Chemometrics Unit 4 Response Surface Methodology Chemometrics Unit 4 Response Surface Methodology Chemometrics Unit 4. Response Surface Methodology In Unit 3 the first two phases of experimental design - definition and screening - were discussed. In

More information

Design and Analysis of Experiments

Design and Analysis of Experiments Design and Analysis of Experiments Part IX: Response Surface Methodology Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com Methods Math Statistics Models/Analyses Response

More information

Response Surface Methodology for the Optimization OF Ethylcellulose Microspheres

Response Surface Methodology for the Optimization OF Ethylcellulose Microspheres International Journal of PharmTech Research CODEN (USA): IJPRIF ISSN : 0974-4304 Vol. 3, No., pp 775-783, April-June 011 Response Surface Methodology for the Optimization OF Ethylcellulose Microspheres

More information

Useful Numerical Statistics of Some Response Surface Methodology Designs

Useful Numerical Statistics of Some Response Surface Methodology Designs Journal of Mathematics Research; Vol. 8, No. 4; August 20 ISSN 19-9795 E-ISSN 19-9809 Published by Canadian Center of Science and Education Useful Numerical Statistics of Some Response Surface Methodology

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

2 Introduction to Response Surface Methodology

2 Introduction to Response Surface Methodology 2 Introduction to Response Surface Methodology 2.1 Goals of Response Surface Methods The experimenter is often interested in 1. Finding a suitable approximating function for the purpose of predicting a

More information

Check boxes of Edited Copy of Sp Topics (was 217-pilot)

Check boxes of Edited Copy of Sp Topics (was 217-pilot) Check boxes of Edited Copy of 10024 Sp 11 213 Topics (was 217-pilot) College Algebra, 9th Ed. [open all close all] R-Basic Algebra Operations Section R.1 Integers and rational numbers Rational and irrational

More information

Unit 12: Response Surface Methodology and Optimality Criteria

Unit 12: Response Surface Methodology and Optimality Criteria Unit 12: Response Surface Methodology and Optimality Criteria STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Revisit your knowledge of polynomial regression Know how to use

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

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

2 k, 2 k r and 2 k-p Factorial Designs

2 k, 2 k r and 2 k-p Factorial Designs 2 k, 2 k r and 2 k-p Factorial Designs 1 Types of Experimental Designs! Full Factorial Design: " Uses all possible combinations of all levels of all factors. n=3*2*2=12 Too costly! 2 Types of Experimental

More information

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,500 108,000 1.7 M Open access books available International authors and editors Downloads Our

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

REVIEW OF EXPERIMENTAL DESIGN IN ANALYTICAL CHEMISTRY

REVIEW OF EXPERIMENTAL DESIGN IN ANALYTICAL CHEMISTRY Page55 Indo American Journal of Pharmaceutical Research, 217 ISSN NO: 2231-6876 REVIEW OF EXPERIMENTAL DESIGN IN ANALYTICAL CHEMISTRY T.Sudha *, G.Divya, J.Sujaritha, P. Duraimurugan Department of Pharmaceutical

More information

Design of Engineering Experiments Part 5 The 2 k Factorial Design

Design of Engineering Experiments Part 5 The 2 k Factorial Design Design of Engineering Experiments Part 5 The 2 k Factorial Design Text reference, Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high

More information

CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES

CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES 6.1 Introduction It has been found from the literature review that not much research has taken place in the area of machining of carbon silicon

More information

The integration of response surface method in microsoft excel with visual basic application

The integration of response surface method in microsoft excel with visual basic application Journal of Physics: Conference Series PAPER OPEN ACCESS The integration of response surface method in microsoft excel with visual basic application To cite this article: H Sofyan et al 2018 J. Phys.: Conf.

More information

ON THE DESIGN POINTS FOR A ROTATABLE ORTHOGONAL CENTRAL COMPOSITE DESIGN

ON THE DESIGN POINTS FOR A ROTATABLE ORTHOGONAL CENTRAL COMPOSITE DESIGN ON THE DESIGN POINTS FOR A ROTATABLE ORTHOGONAL CENTRAL COMPOSITE DESIGN Authors: CHRISTOS P. KITSOS Department of Informatics, Technological Educational Institute of Athens, Greece (xkitsos@teiath.gr)

More information

Searching for D-E cient Equivalent-Estimation Second-Order Split-Plot Designs

Searching for D-E cient Equivalent-Estimation Second-Order Split-Plot Designs Searching for D-E cient Equivalent-Estimation Second-Order Split-Plot Designs NAM-KY NGUYEN VIASM and VNU International School, Hanoi, Vietnam TUNG-DINH PHAM VNU University of Science, Hanoi, Vietnam Several

More information

Experimental designs for multiple responses with different models

Experimental designs for multiple responses with different models Graduate Theses and Dissertations Graduate College 2015 Experimental designs for multiple responses with different models Wilmina Mary Marget Iowa State University Follow this and additional works at:

More information

ALGEBRA 2. Background Knowledge/Prior Skills Knows what operation properties hold for operations with matrices

ALGEBRA 2. Background Knowledge/Prior Skills Knows what operation properties hold for operations with matrices ALGEBRA 2 Numbers and Operations Standard: 1 Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, and number

More information

Hardware/Software Design Methodologies Introduction to DoE and Course Projects

Hardware/Software Design Methodologies Introduction to DoE and Course Projects Hardware/Software Design Methodologies Introduction to DoE and Course Projects Vittorio Zaccaria Dipartimento di Elettronica e Informazione Politecnico di Milano zaccaria@elet.polimi.it for informations

More information

AN ALTERNATIVE APPROACH TO EVALUATION OF POOLABILITY FOR STABILITY STUDIES

AN ALTERNATIVE APPROACH TO EVALUATION OF POOLABILITY FOR STABILITY STUDIES Journal of Biopharmaceutical Statistics, 16: 1 14, 2006 Copyright Taylor & Francis, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543400500406421 AN ALTERNATIVE APPROACH TO EVALUATION OF POOLABILITY

More information

Available online Research Article

Available online   Research Article Available online www.jocpr.com Journal of Chemical and Pharmaceutical Research, 2013, 5(12):454-458 Research Article ISSN : 0975-7384 CODEN(USA) : JCPRC5 Optimization of microwave-assisted extraction of

More information

OPTIMIZATION OF FIRST ORDER MODELS

OPTIMIZATION OF FIRST ORDER MODELS Chapter 2 OPTIMIZATION OF FIRST ORDER MODELS One should not multiply explanations and causes unless it is strictly necessary William of Bakersville in Umberto Eco s In the Name of the Rose 1 In Response

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

Disegno Sperimentale (DoE) come strumento per QbD

Disegno Sperimentale (DoE) come strumento per QbD Giornata di Studio Disegno Sperimentale (DoE) come strumento per QbD Università degli Studi di Milano Dipartimento di Scienze Farmaceutiche Milano, 22 aprile 2013 Dr. Lorenza Broccardo Introduction Nowadays,

More information

SOME APPLICATIONS OF STATISTICAL DESIGN OF EXPERIMENT METHODOLOGY IN CIVIL ENGINEERING

SOME APPLICATIONS OF STATISTICAL DESIGN OF EXPERIMENT METHODOLOGY IN CIVIL ENGINEERING Congrès annuel de la Société canadienne de génie civil Annual Conference of the Canadian Society for Civil Engineering Moncton, Nouveau-Brunswick, Canada 4-7 juin 2003 / June 4-7, 2003 SOME APPLICATIONS

More information

CONTROLLED SEQUENTIAL FACTORIAL DESIGN FOR SIMULATION FACTOR SCREENING. Hua Shen Hong Wan

CONTROLLED SEQUENTIAL FACTORIAL DESIGN FOR SIMULATION FACTOR SCREENING. Hua Shen Hong Wan Proceedings of the 2005 Winter Simulation Conference M. E. Kuhl, N. M. Steiger, F. B. Armstrong, and J. A. Joines, eds. CONTROLLED SEQUENTIAL FACTORIAL DESIGN FOR SIMULATION FACTOR SCREENING Hua Shen Hong

More information

14.0 RESPONSE SURFACE METHODOLOGY (RSM)

14.0 RESPONSE SURFACE METHODOLOGY (RSM) 4. RESPONSE SURFACE METHODOLOGY (RSM) (Updated Spring ) So far, we ve focused on experiments that: Identify a few important variables from a large set of candidate variables, i.e., a screening experiment.

More information

Formative Assignment PART A

Formative Assignment PART A MHF4U_2011: Advanced Functions, Grade 12, University Preparation Unit 2: Advanced Polynomial and Rational Functions Activity 2: Families of polynomial functions Formative Assignment PART A For each of

More information

Classes of Second-Order Split-Plot Designs

Classes of Second-Order Split-Plot Designs Classes of Second-Order Split-Plot Designs DATAWorks 2018 Springfield, VA Luis A. Cortés, Ph.D. The MITRE Corporation James R. Simpson, Ph.D. JK Analytics, Inc. Peter Parker, Ph.D. NASA 22 March 2018 Outline

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Definitive Screening Designs with Added Two-Level Categorical Factors *

Definitive Screening Designs with Added Two-Level Categorical Factors * Definitive Screening Designs with Added Two-Level Categorical Factors * BRADLEY JONES SAS Institute, Cary, NC 27513 CHRISTOPHER J NACHTSHEIM Carlson School of Management, University of Minnesota, Minneapolis,

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

STUDY OF THE APPLICABILTY OF CONTENT UNIFORMITY AND DISSOLUTION VARIATION TEST ON ROPINIROLE HYDROCHLORIDE TABLETS

STUDY OF THE APPLICABILTY OF CONTENT UNIFORMITY AND DISSOLUTION VARIATION TEST ON ROPINIROLE HYDROCHLORIDE TABLETS & STUDY OF THE APPLICABILTY OF CONTENT UNIFORMITY AND DISSOLUTION VARIATION TEST ON ROPINIROLE HYDROCHLORIDE TABLETS Edina Vranić¹*, Alija Uzunović² ¹ Department of Pharmaceutical Technology, Faculty of

More information

Check boxes of Edited Copy of Sp Topics (was 261-pilot)

Check boxes of Edited Copy of Sp Topics (was 261-pilot) Check boxes of Edited Copy of 10023 Sp 11 253 Topics (was 261-pilot) Intermediate Algebra (2011), 3rd Ed. [open all close all] R-Review of Basic Algebraic Concepts Section R.2 Ordering integers Plotting

More information

PRODUCT QUALITY IMPROVEMENT THROUGH RESPONSE SURFACE METHODOLOGY : A CASE STUDY

PRODUCT QUALITY IMPROVEMENT THROUGH RESPONSE SURFACE METHODOLOGY : A CASE STUDY PRODUCT QULITY IMPROVEMENT THROUGH RESPONSE SURFCE METHODOLOGY : CSE STUDY HE Zhen, College of Management and Economics, Tianjin University, China, zhhe@tju.edu.cn, Tel: +86-22-8740783 ZHNG Xu-tao, College

More information

Application of mathematical, statistical, graphical or symbolic methods to maximize chemical information.

Application of mathematical, statistical, graphical or symbolic methods to maximize chemical information. Application of mathematical, statistical, graphical or symbolic methods to maximize chemical information. -However, this definition can be expanded to include: biology (biometrics), environmental science

More information

Algebra 2. Curriculum (524 topics additional topics)

Algebra 2. Curriculum (524 topics additional topics) Algebra 2 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

More information

Chapter 1. Gaining Knowledge with Design of Experiments

Chapter 1. Gaining Knowledge with Design of Experiments Chapter 1 Gaining Knowledge with Design of Experiments 1.1 Introduction 2 1.2 The Process of Knowledge Acquisition 2 1.2.1 Choosing the Experimental Method 5 1.2.2 Analyzing the Results 5 1.2.3 Progressively

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

Experimental designs for precise parameter estimation for non-linear models

Experimental designs for precise parameter estimation for non-linear models Minerals Engineering 17 (2004) 431 436 This article is also available online at: www.elsevier.com/locate/mineng Experimental designs for precise parameter estimation for non-linear models Z. Xiao a, *,

More information

Experimental Space-Filling Designs For Complicated Simulation Outpts

Experimental Space-Filling Designs For Complicated Simulation Outpts Experimental Space-Filling Designs For Complicated Simulation Outpts LTC Alex MacCalman PhD Student Candidate Modeling, Virtual Environments, and Simulations (MOVES) Institute Naval Postgraduate School

More information

Objective Experiments Glossary of Statistical Terms

Objective Experiments Glossary of Statistical Terms Objective Experiments Glossary of Statistical Terms This glossary is intended to provide friendly definitions for terms used commonly in engineering and science. It is not intended to be absolutely precise.

More information

USA Mathematical Talent Search Round 1 Solutions Year 27 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 27 Academic Year 1/1/27. Fill in the spaces of the grid to the right with positive integers so that in each 2 2 square with top left number a, top right number b, bottom left number c, and bottom right number d, either

More information

Unit 8 - Polynomial and Rational Functions Classwork

Unit 8 - Polynomial and Rational Functions Classwork Unit 8 - Polynomial and Rational Functions Classwork This unit begins with a study of polynomial functions. Polynomials are in the form: f ( x) = a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 2 x 2 + a

More information

MULTIPLE REGRESSION METHODS

MULTIPLE REGRESSION METHODS DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 816 MULTIPLE REGRESSION METHODS I. AGENDA: A. Residuals B. Transformations 1. A useful procedure for making transformations C. Reading:

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

When combining power functions into a single polynomial function, there are a few new features we like to look for, such as

When combining power functions into a single polynomial function, there are a few new features we like to look for, such as Section 1.2 Characteristics of Polynomial Functions In section 1.1 we explored Power Functions, a single piece of a polynomial function. This modelling method works perfectly for simple real world problems

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

College Algebra and Trigonometry

College Algebra and Trigonometry GLOBAL EDITION College Algebra and Trigonometry THIRD EDITION J. S. Ratti Marcus McWaters College Algebra and Trigonometry, Global Edition Table of Contents Cover Title Page Contents Preface Resources

More information

Check boxes of Edited Copy of Sp Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and

Check boxes of Edited Copy of Sp Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Check boxes of Edited Copy of 10021 Sp 11 152 Topics (was 145 for pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Additional Topics Appendix Course Readiness Multiplication

More information

Project: PAJ3: Combined Cyclic Loading and Hostile Environments Report 3. A Guide to the use of Design of Experiment Methods

Project: PAJ3: Combined Cyclic Loading and Hostile Environments Report 3. A Guide to the use of Design of Experiment Methods Project: PAJ3: Combined Cyclic Loading and Hostile Environments Report 3 A Guide to the use of Design of Experiment Methods A OLUSANYA M HALL, XYRATEX, HAVANT May 1997 A Guide to the use of Design of Experiment

More information

RESPONSE SURFACE METHODOLOGY

RESPONSE SURFACE METHODOLOGY RESPONSE SURFACE METHODOLOGY WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain

More information

DESK Secondary Math II

DESK Secondary Math II Mathematical Practices The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically

More information

Definitive Screening Designs

Definitive Screening Designs Definitive Screening Designs Bradley Jones September 2011 Copyright 2008, SAS Institute Inc. All rights reserved. Joint work with Chris Nachtsheim Outline 1. Motivation 2. Design Structure 3. Design Construction

More information

Taguchi Method and Robust Design: Tutorial and Guideline

Taguchi Method and Robust Design: Tutorial and Guideline Taguchi Method and Robust Design: Tutorial and Guideline CONTENT 1. Introduction 2. Microsoft Excel: graphing 3. Microsoft Excel: Regression 4. Microsoft Excel: Variance analysis 5. Robust Design: An Example

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

Binomials defined, 13 division by, FOIL method and, 22 multiplying binomial by trinomial,

Binomials defined, 13 division by, FOIL method and, 22 multiplying binomial by trinomial, 5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-1 Index Absolute value defined, 46 functions and, 126 59 Absolute value equations, solutions, 46 49 Absolute value inequalities, solutions, 263 267 Acceleration,

More information

9th and 10th Grade Math Proficiency Objectives Strand One: Number Sense and Operations

9th and 10th Grade Math Proficiency Objectives Strand One: Number Sense and Operations Strand One: Number Sense and Operations Concept 1: Number Sense Understand and apply numbers, ways of representing numbers, the relationships among numbers, and different number systems. Justify with examples

More information

FIVE-FACTOR CENTRAL COMPOSITE DESIGNS ROBUST TO A PAIR OF MISSING OBSERVATIONS. Munir Akhtar Vice-Chanselor, Islamia University Bahawalpur, Pakistan.

FIVE-FACTOR CENTRAL COMPOSITE DESIGNS ROBUST TO A PAIR OF MISSING OBSERVATIONS. Munir Akhtar Vice-Chanselor, Islamia University Bahawalpur, Pakistan. Journal of Research (Science), Bahauddin Zakariya University, Multan, Pakistan. Vol.2, No.2, December 2, pp. 5-5 ISSN 2-2 FIVE-FACTOR CENTRAL COMPOSITE DESIGNS ROBUST TO A PAIR OF MISSING OBSERVATIONS

More information

Process/product optimization using design of experiments and response surface methodology

Process/product optimization using design of experiments and response surface methodology Process/product optimization using design of experiments and response surface methodology Mikko Mäkelä Sveriges landbruksuniversitet Swedish University of Agricultural Sciences Department of Forest Biomaterials

More information

MATHEMATICS SYLLABUS SECONDARY 4th YEAR

MATHEMATICS SYLLABUS SECONDARY 4th YEAR European Schools Office of the Secretary-General Pedagogical Development Unit Ref.:010-D-591-en- Orig.: EN MATHEMATICS SYLLABUS SECONDARY 4th YEAR 6 period/week course APPROVED BY THE JOINT TEACHING COMMITTEE

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

Algebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target

Algebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target Algebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target SB Activity Activity 1 Investigating Patterns 1-1 Learning Targets: Identify patterns in data. Use tables, graphs,

More information

AN ENHANCED RECURSIVE STOPPING RULE FOR STEEPEST ASCENT SEARCHES IN RESPONSE SURFACE METHODOLOGY.

AN ENHANCED RECURSIVE STOPPING RULE FOR STEEPEST ASCENT SEARCHES IN RESPONSE SURFACE METHODOLOGY. AN ENHANCED RECURSIVE STOPPING RULE FOR STEEPEST ASCENT SEARCHES IN RESPONSE SURFACE METHODOLOGY. Guillermo Miró and Enrique Del Castillo Department of Industrial and Manufacturing Engineering The Pennsylvania

More information

Eighth Grade Algebra I Mathematics

Eighth Grade Algebra I Mathematics Description The Appleton Area School District middle school mathematics program provides students opportunities to develop mathematical skills in thinking and applying problem-solving strategies. The framework

More information

ACT MATH MUST-KNOWS Pre-Algebra and Elementary Algebra: 24 questions

ACT MATH MUST-KNOWS Pre-Algebra and Elementary Algebra: 24 questions Pre-Algebra and Elementary Algebra: 24 questions Basic operations using whole numbers, integers, fractions, decimals and percents Natural (Counting) Numbers: 1, 2, 3 Whole Numbers: 0, 1, 2, 3 Integers:

More information

Howard Mark and Jerome Workman Jr.

Howard Mark and Jerome Workman Jr. Linearity in Calibration: How to Test for Non-linearity Previous methods for linearity testing discussed in this series contain certain shortcomings. In this installment, the authors describe a method

More information

Orthogonal, Planned and Unplanned Comparisons

Orthogonal, Planned and Unplanned Comparisons This is a chapter excerpt from Guilford Publications. Data Analysis for Experimental Design, by Richard Gonzalez Copyright 2008. 8 Orthogonal, Planned and Unplanned Comparisons 8.1 Introduction In this

More information

NEW YORK ALGEBRA TABLE OF CONTENTS

NEW YORK ALGEBRA TABLE OF CONTENTS NEW YORK ALGEBRA TABLE OF CONTENTS CHAPTER 1 NUMBER SENSE & OPERATIONS TOPIC A: Number Theory: Properties of Real Numbers {A.N.1} PART 1: Closure...1 PART 2: Commutative Property...2 PART 3: Associative

More information

1 ** The performance objectives highlighted in italics have been identified as core to an Algebra II course.

1 ** The performance objectives highlighted in italics have been identified as core to an Algebra II course. Strand One: Number Sense and Operations Every student should understand and use all concepts and skills from the pervious grade levels. The standards are designed so that new learning builds on preceding

More information

Mixture Designs Based On Hadamard Matrices

Mixture Designs Based On Hadamard Matrices Statistics and Applications {ISSN 2452-7395 (online)} Volume 16 Nos. 2, 2018 (New Series), pp 77-87 Mixture Designs Based On Hadamard Matrices Poonam Singh 1, Vandana Sarin 2 and Rashmi Goel 2 1 Department

More information

2. TRIGONOMETRY 3. COORDINATEGEOMETRY: TWO DIMENSIONS

2. TRIGONOMETRY 3. COORDINATEGEOMETRY: TWO DIMENSIONS 1 TEACHERS RECRUITMENT BOARD, TRIPURA (TRBT) EDUCATION (SCHOOL) DEPARTMENT, GOVT. OF TRIPURA SYLLABUS: MATHEMATICS (MCQs OF 150 MARKS) SELECTION TEST FOR POST GRADUATE TEACHER(STPGT): 2016 1. ALGEBRA Sets:

More information

A Study on Minimal-point Composite Designs

A Study on Minimal-point Composite Designs A Study on Minimal-point Composite Designs by Yin-Jie Huang Advisor Ray-Bing Chen Institute of Statistics, National University of Kaohsiung Kaohsiung, Taiwan 8 R.O.C. July 007 Contents Introduction Construction

More information

Chapter 6 The 2 k Factorial Design Solutions

Chapter 6 The 2 k Factorial Design Solutions Solutions from Montgomery, D. C. (004) Design and Analysis of Experiments, Wiley, NY Chapter 6 The k Factorial Design Solutions 6.. A router is used to cut locating notches on a printed circuit board.

More information

Course of Study For Math

Course of Study For Math Medina County Schools Course of Study For Math Algebra II (Cloverleaf) Algebra II Honors (Cloverleaf) 321 STANDARD 1: Number, Number Sense and Operations Students demonstrate number sense, including an

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

Uncertainty of the Measurement of Radial Runout, Axial Runout and Coning using an Industrial Axi-Symmetric Measurement Machine

Uncertainty of the Measurement of Radial Runout, Axial Runout and Coning using an Industrial Axi-Symmetric Measurement Machine 3924: 38 th MATADOR Conference Uncertainty of the Measurement of Radial Runout, Axial Runout and Coning using an Industrial Axi-Symmetric Measurement Machine J E Muelaner, A Francis, P G Maropoulos The

More information

1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to

1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic

More information

Practice of SAS Logistic Regression on Binary Pharmacodynamic Data Problems and Solutions. Alan J Xiao, Cognigen Corporation, Buffalo NY

Practice of SAS Logistic Regression on Binary Pharmacodynamic Data Problems and Solutions. Alan J Xiao, Cognigen Corporation, Buffalo NY Practice of SAS Logistic Regression on Binary Pharmacodynamic Data Problems and Solutions Alan J Xiao, Cognigen Corporation, Buffalo NY ABSTRACT Logistic regression has been widely applied to population

More information

MATH 118 FINAL EXAM STUDY GUIDE

MATH 118 FINAL EXAM STUDY GUIDE MATH 118 FINAL EXAM STUDY GUIDE Recommendations: 1. Take the Final Practice Exam and take note of questions 2. Use this study guide as you take the tests and cross off what you know well 3. Take the Practice

More information

Contents. 2 2 factorial design 4

Contents. 2 2 factorial design 4 Contents TAMS38 - Lecture 10 Response surface methodology Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics 12 December, 2017 2 2 factorial design Polynomial Regression model First

More information

Addition of Center Points to a 2 k Designs Section 6-6 page 271

Addition of Center Points to a 2 k Designs Section 6-6 page 271 to a 2 k Designs Section 6-6 page 271 Based on the idea of replicating some of the runs in a factorial design 2 level designs assume linearity. If interaction terms are added to model some curvature results

More information

Sect Polynomial and Rational Inequalities

Sect Polynomial and Rational Inequalities 158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax

More information

Randomisation, Replication, Response Surfaces. and. Rosemary

Randomisation, Replication, Response Surfaces. and. Rosemary Randomisation, Replication, Response Surfaces and Rosemary 1 A.C. Atkinson a.c.atkinson@lse.ac.uk Department of Statistics London School of Economics London WC2A 2AE, UK One joint publication RAB AND ME

More information

Two-Stage Computing Budget Allocation. Approach for Response Surface Method PENG JI

Two-Stage Computing Budget Allocation. Approach for Response Surface Method PENG JI Two-Stage Computing Budget Allocation Approach for Response Surface Method PENG JI NATIONAL UNIVERSITY OF SINGAPORE 2005 Two-Stage Computing Budget Allocation Approach for Response Surface Method PENG

More information

by Jerald Murdock, Ellen Kamischke, and Eric Kamischke An Investigative Approach

by Jerald Murdock, Ellen Kamischke, and Eric Kamischke An Investigative Approach Number and Operations Understand numbers, ways of representing numbers, relationships among numbers, and number systems develop a deeper understanding of very large and very small numbers and of various

More information