Concrete Mixture Optimization Using 

Statistical Methods: Final Report 

FHWA-RD-03-060 

M.J. Simon 

FHWA Office of Infrastructure Research and Development 

6300 Georgetown Pike 

McLean, VA 22101 

National Institute of Standards and Technology, Building Materials Division 

100 Bureau Drive 

Gaithersburg, MD 20899 

National Institute of Standards and Technology, Statistical Engineering Division 

100 Bureau Drive 

Gaithersburg, MD 20899
FOREWORD 

This report presents the results of a study conducted jointly by the Federal Highway 

Administration and the National Institute of Standards and Technology to assess the feasibility of 

using statistical experiment design and analysis methods to optimize concrete mixture 

proportions. The laboratory phase of the study indicated that both the classical mixture method 

and the factorial approach could be applied to the problem of optimizing concrete mixture 

proportions. The factorial approach was used as the basis for developing an Internet-based 

computer program, the Concrete Optimization Software Tool, in the second phase of this project. 

This tool, accessible on the Web, allows a potential user to learn about and try this statistical 

approach. This report will be of interest to materials engineers and others who are involved in 

concrete construction and concrete mixture design, materials selection, and proportioning. 

 T. Paul Teng, P.E. 

 Director, Office of Infrastructure 

 Research and Development 

NOTICE 

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

in the interest of information exchange. The U.S. Government assumes no liability for its 

contents or use thereof. This report does not constitute a standard, specification, or regulation. 

The U.S. Government does not endorse products or manufacturers. Trade and manufacturers’ 

names appear in this report only because they are considered essential to the object of the 

document. 
Technical Report Documentation Page 

1. Report No.

FHWA-RD-03-060 

2. Government Accession No. 

 

3. Recipient’s Catalog No.

 

5. Report Date 

 

4. Title and Subtitle 

Concrete Mixture Optimization Using Statistical Methods: Final Report 

6. Performing Organization Code

480017 

7. Author(s) 

M.J. Simon 

8. Performing Organization Report No. 

 

10. Work Unit No. 

 

9. Performing Organization Name and Address 

FHWA Office of Infrastructure Research and Development, 6300 

Georgetown Pike, McLean, VA 22101 

National Institute of Standards and Technology, Building Materials 

Division, 100 Bureau Drive, Gaithersburg, MD 20899 

National Institute of Standards and Technology, Statistical Engineering 

Division, 100 Bureau Drive, Gaithersburg, MD 20899 

 

11. Contract or Grant No. 

DTFH61-97-Y-30033 

13. Type of Report and Period Covered 

Final Report 

November, 1996 to July, 2003 

12. Sponsoring Agency Name and Address 

Office of Infrastructure Research and Development 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296 

14. Sponsoring Agency Code 

 

15. Supplementary Notes 

This project was a cooperative effort between the FWHA Office of Infrastructure Research and Development, the NIST Building 

and Fire Reseach Laboratory (Building Materials Division), and the NIST Information Technology Laboratory (Statistical 

Engineering Division). 

16. Abstract 

This report presents the results of a research project whose goals were to investigate the feasibility of using statistical experiment 

design and analysis methods to optimize concrete mixture proportions and to develop an Internet-based software program to 

optimize concrete mixtures using these methods. Two experiment design approaches (classical mixture and factorial-based 

central composite design) were investigated in laboratory experiments. In each case, six component materials were used, and 

mixtures were optimized for four performance criteria (properties) and cost. Based on the experimental results, the factorialbased approach was selected as the basis for the Internet-based system. This system, the Concrete Optimization Software Tool 

(COST), employs a six-step interactive procedure starting with materials selection and working through trial batches, testing, and 

analysis of test results. The end result is recommended mixture proportions to achieve the desired performance levels. COST 

was developed as a tool to introduce the industry to the potential benefits of using statistical methods in concrete mixture 

proportioning, and to give interested parties an opportunity to try the methods for themselves. 

17. Key Words 

 

Building technology, pavements, structures, concrete, 

mixture proportioning, experiment design, optimization, 

response surface methods 

18. Distribution Statement 

No restrictions. This document is available to the public through 

the National Technical Information Service, Springfield, VA 

22161. 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No of Pages

167 

22. Price 

 

 Form DOT F 1700.7 (8-72) Reproduction of completed pages authorized 
ii

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH 

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha 

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS 

oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius oC 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx 

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH

mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2

 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac 

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for th International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E e 380. 

(Revised March 2003)
iii

 TABLE OF CONTENTS 

CHAPTER 1. Introduction ..........................................................................................................1 

1.1 Statement of Problem and Project Goals ..........................................................................1 

1.2 Scope of Report.................................................................................................................3 

CHAPTER 2. Background on Statistical Methods....................................................................5 

2.1 Response Surface Methodology .......................................................................................5 

2.2 Experiment Design............................................................................................................5 

2.2.1 Mixture Approach.................................................................................................5 

2.2.2 Factorial (MIV) Approach ....................................................................................8 

2.3 Model Fitting and Validation..........................................................................................10 

2.4 Optimization ...................................................................................................................13 

CHAPTER 3. Laboratory Experiment Using Mixture Approach .........................................15 

3.1 Introduction.....................................................................................................................15 

3.2 Selection of Materials, Proportions, and Constraints......................................................15 

3.3 Experiment Design Details .............................................................................................15 

3.4 Specimen Fabrication and Testing..................................................................................16 

3.5 Results and Analysis.......................................................................................................18 

3.5.1 Measured Responses...........................................................................................18 

3.5.2 Model Identification and Validation for 28-Day Strength..................................18 

3.5.3 Models for Other Responses...............................................................................21 

3.6 Optimization ...................................................................................................................21 

3.6.1 Graphical Optimization.......................................................................................21 

3.6.2 Numerical Optimization......................................................................................25 

3.6.3 Accounting for Uncertainty ................................................................................26 

CHAPTER 4. Laboratory Experiment Using Factorial Approach........................................27 

4.1 Introduction.....................................................................................................................27 

4.2 Selection of Materials, Proportions, and Constraints......................................................27 

4.3 Experiment Design Details .............................................................................................28 

4.4 Specimen Fabrication and Testing..................................................................................28 

4.5 Results and Analysis.......................................................................................................30 

4.5.1 Responses............................................................................................................30 

4.5.2 Exploratory Data Analysis for 1-Day Strength...................................................30 

4.5.3 Model Fitting and Validation for 1-Day Strength...............................................34 

4.5.4 Models for Other Responses...............................................................................37 

4.6 Optimization ...................................................................................................................37 

4.6.1 Graphical Optimization.......................................................................................38 

4.6.2 Numerical Optimization......................................................................................40 

4.6.3 Accounting for Uncertainty ................................................................................41 
iv

TABLE OF CONTENTS (continued) 

CHAPTER 5. Development of Interactive Web Site (COST Program) ................................43 

5.1 Introduction.....................................................................................................................43 

5.2 Selection of Approach ....................................................................................................43 

5.3 Considerations in Development......................................................................................44 

5.4 Description of the Software and Web Site......................................................................45 

 5.4.1 Introduction.........................................................................................................45

 5.4.2 Overview of COST Six-Step Process .................................................................46 

5.5 Future Considerations .....................................................................................................50 

ACKNOWLEDGMENTS ...........................................................................................................51 

REFERENCES.............................................................................................................................53 

APPENDIX A. Experiment Design and Data Analysis for Mixture Experiment............... A-1 

A.1 Experiment Design and Response Data....................................................................... A-1 

A.2 Data Analysis and Model Fitting ................................................................................. A-4 

 A.2.1 Slump ................................................................................................................ A-4 

 A.2.2 1-Day Strength .................................................................................................. A-9 

 A.2.3 28-Day Strength .............................................................................................. A-15 

 A.2.4 RCT Charge Passed ........................................................................................ A-21 

APPENDIX B. Experiment Design and Data Analysis for Factorial Experiment ..............B-1 

B.1 Experiment Design and Response Data........................................................................B-1 

B.2 Data Analysis and Model Fitting ..................................................................................B-4 

 B.2.1 Slump .................................................................................................................B-4 

 B.2.2 1-Day Strength .................................................................................................B-11 

 B.2.3 28-Day Strength ...............................................................................................B-18 

 B.2.4 RCT Charge Passed (coulombs) ......................................................................B-25 

APPENDIX C. COST User’s Guide....................................................................................... C-1 

Abstract..................................................................................................................................C-2 

Section 1. Overview...............................................................................................................C-3 

C1.1 Introduction...................................................................................................................C-3 

C1.2 Scope.............................................................................................................................C-4 

C1.3 System Requirements....................................................................................................C-4

C1.4 Disclaimer.....................................................................................................................C-5 

C1.5 General Information......................................................................................................C-5 

 C1.5.1 COST Homepage and Main Menu...................................................................C-5 

Section 2. Using COST..........................................................................................................C-9 

C2.1 Background and Preliminary Planning .........................................................................C-9 

 C2.1.1 Responses.........................................................................................................C-9 

 C2.1.2 Factors ............................................................................................................C-10 

C2.2 Step 1—Specify Responses........................................................................................C-12 

C2.3 Step 2—Specify Measures .........................................................................................C-14 
v

 C2.3.1 Instructions for Section 1: Number of Parameters (Factors) to Vary ............C-14 

 C2.3.2 Instructions for Section 2: Select w/c or w/cm ..............................................C-14 

 C2.3.3 Instructions for Section 3: Select Other Mixture Components ......................C-15 

 C2.3.4 Specific Instructions for Mineral Admixtures................................................C-16 

 C2.3.5 Specific Instructions for Chemical Admixtures.............................................C-16 

 C2.3.6 Specific Instructions for Aggregates..............................................................C-17 

 C2.3.7 Instructions for Section 4: Additional Information........................................C-17 

C2.4 Step 3—Run Trial Batches.........................................................................................C-20 

 C2.4.1 Guidelines for Running Trial Batches............................................................C-20 

 C2.4.2 Nuisance Factors and Run Sequence Randomization....................................C-21 

 C2.4.3 Running the Experiment.................................................................................C-22 

C2.5 Step 4—Input Results.................................................................................................C-23

 C2.5.1 Instructions for Changing Cost Information ..................................................C-23 

 C2.5.2 Instructions for Entering/Editing Data ...........................................................C-23 

C2.6 Step 5—Analyze Data................................................................................................C-25 

 C2.6.1 Instructions for Changing Response Information ..........................................C-25 

 C2.6.2 Analysis Tasks................................................................................................C-25 

C2.7 Step 6—Summarize Analysis.....................................................................................C-39 

Section 3. References...........................................................................................................C-41 
vi

LIST OF FIGURES 

Figure Page

 1 Example of triangular simplex region from three-component mixture experiment .........6 

 2 Simplex-centroid design for three variables .....................................................................6 

 3 Example of subregion of full simplex containing range of feasible mixtures ..................8 

 4 Schematic of a central composite design for three factors ...............................................9 

 5 Examples of desirability functions..................................................................................14 

 6 Response trace plot for 28-day strength (mixture experiment) ......................................22 

 7 Contour plot for 28-day strength in water, cement, and HRWRA .................................23 

 8 Contour plot for 28-day strength in water, cement, and silica fume...............................23 

 9 Contour plot for 28-day strength in water, coarse aggregate, and fine aggregate ..........24 

 10 Contour plot for 28-day strength in silica fume, coarse aggregate, and fine aggregate .24 

 11 Desirability functions for responses in mixture experiment...........................................25 

 12 Raw data plot for 1-day strength (factorial experiment).................................................32 

 13 Scatterplot showing effect of w/c on 1-day strength (factorial experiment) ..................32 

 14 Means plot for 1-day strength (factorial experiment).....................................................33 

 15 Example of cube plot for factorial points .......................................................................33 

 16 Example of a normal probability plot for model validation ...........................................36 

 17 Example of a residual plot (residuals vs. run) for model validation...............................37 

 18 28-day strength in w/c and silica fume (HRWRA at middle setting).............................38 

 19 28-day strength in w/c and silica fume (HRWRA at high setting).................................39 

 20 Overlay plot for RCT < 700 and slump = 50–100 mm...................................................39 

 21 Overlay plot for RCT < 700, slump 50–100 mm, and 28-day strength > 51 MPa .........40 

 22 Desirability functions for factorial experiment...............................................................40 

 23 Summary screen from COST..........................................................................................49 

 A-1 Mixture experiment: normal probability plot for slump............................................. A-6 

 A-2 Mixture experiment: residuals vs. run for slump........................................................ A-6 

 A-3 Mixture experiment: Cook’s distance for slump ........................................................ A-7 

 A-4 Mixture experiment: trace plot for slump ................................................................... A-7 

 A-5 Mixture experiment: contour plot for slump in water, cement, and silica fume......... A-8 

 A-6 Mixture experiment: contour plot of slump in water, cement, and HRWRA............. A-8 

 A-7 Mixture experiment: normal probability plot for 1-day strength.............................. A-11 

 A-8 Mixture experiment: residuals vs. run for 1-day strength ........................................ A-11 

 A-9 Mixture experiment: trace plot for 1-day strength.................................................... A-12 

A-10 Mixture experiment: contour plot of 1-day strength in water, cement, and 

 silica fume.................................................................................................................. A-12 

A-11 Mixture experiment: contour plot of 1-day strength in water, cement, and 

 HRWRA..................................................................................................................... A-13 

A-12 Mixture experiment: contour plot of 1-day strength in water, cement, and fine 

 aggregate.................................................................................................................... A-13 

A-13 Mixture experiment: contour plot of 1-day strength in silica fume, HRWRA, 

 and fine aggregate...................................................................................................... A-14 

A-14 Mixture experiment: contour plot of 1-day strength in silica fume, coarse aggregate, 

 and fine aggregate...................................................................................................... A-14 
vii

LIST OF FIGURES (continued) 

Figure Page

A-15 Mixture experiment: normal probability plot for 28-day strength............................ A-17 

A-16 Mixture experiment: residuals vs. run for 28-day strength....................................... A-17 

A-17 Mixture experiment: Cook’s distance vs. run for 28-day strength ........................... A-18 

A-18 Mixture experiment: trace plot for 28-day strength.................................................. A-18 

A-19 Mixture experiment: contour plot of 28-day strength in water, silica fume, and 

 HRWRA..................................................................................................................... A-19 

A-20 Mixture experiment: contour plot of 28-day strength in water, silica fume, and 

 coarse aggregate......................................................................................................... A-19

A-21 Mixture experiment: contour plot of 28-day strength in cement, coarse aggregate, 

 and fine aggregate...................................................................................................... A-20 

A-22 Mixture experiment: normal probability plot for RCT charge passed 

(no transform) ............................................................................................................ A-23 

A-23 Mixture experiment: normal probability plot for RCT charge passed 

(natural log transform) ............................................................................................... A-23 

A-24 Mixture experiment: residuals vs. predicted for RCT charge passed 

(no transform) ............................................................................................................ A-24 

A-25 Mixture experiment: residuals vs. run for RCT charge passed (no transform) ........ A-24 

A-26 Mixture experiment: residuals vs. predicted for RCT charge passed 

(natural log transform) ............................................................................................... A-25 

A-27 Mixture experiment: residuals vs. run for RCT charge passed 

(natural log transform) ............................................................................................... A-25 

A-28 Mixture experiment: Cook’s distance for RCT charge passed 

(natural log transform) ............................................................................................... A-26 

A-29 Mixture experiment: trace plot for RCT charge passed (natural log transform) ...... A-26 

A-30 Mixture experiment: contour plot of ln (RCT charge passed) in water, silica fume, 

and coarse aggregate.................................................................................................. A-27 

A-31 Mixture experiment: contour plot of ln (RCT charge passed) in water, silica fume, 

and HRWRA.............................................................................................................. A-27 

A-32 Mixture experiment: contour plot of ln (RCT charge passed) in cement, HRWRA, 

and fine aggregate...................................................................................................... A-28 

 B-1 Factorial experiment: normal probability plot for slump.............................................B-6 

 B-2 Factorial experiment: raw data plot for slump.............................................................B-6 

 B-3 Factorial experiment: scatterplot of slump vs. w/c ......................................................B-7 

 B-4 Factorial experiment: scatterplot of slump vs. coarse aggregate .................................B-7 

 B-5 Factorial experiment: scatterplot of slump vs. fine aggregate .....................................B-8 

 B-6 Factorial experiment: scatterplot of slump vs. HRWRA.............................................B-8 

 B-7 Factorial experiment: scatterplot of slump vs. silica fume ..........................................B-9 

 B-8 Factorial experiment: means plots for slump...............................................................B-9 

 B-9 Factorial experiment: slump vs. run sequence...........................................................B-10 

B-10 Factorial experiment: lag plot for slump....................................................................B-10 

 
viii

LIST OF FIGURES (continued) 

Figure Page 

B-11 Factorial experiment: normal probability plot for 1-day strength .............................B-13 

B-12 Factorial experiment: raw data plot for 1-day strength..............................................B-13 

B-13 Factorial experiment: scatterplot of 1-day strength vs. w/c.......................................B-14 

B-14 Factorial experiment: scatterplot of 1-day strength vs. fine aggregate .....................B-14 

B-15 Factorial experiment: scatterplot of 1-day strength vs. coarse aggregate..................B-15 

B-16 Factorial experiment: scatterplot of 1-day strength vs. HRWRA..............................B-15 

B-17 Factorial experiment: scatterplot of 1-day strength vs. silica fume...........................B-16 

B-18 Factorial experiment: means plot for 1-day strength .................................................B-16 

B-19 Factorial experiment: 1-day strength vs. run sequence..............................................B-17 

B-20 Factorial experiment: lag plot for 1-day strength ......................................................B-17 

B-21 Factorial experiment: normal probability plot for 28-day strength ...........................B-20 

B-22 Factorial experiment: raw data plot for 28-day strength............................................B-20 

B-23 Factorial experiment: scatterplot of 28-day strength vs. w/c.....................................B-21 

B-24 Factorial experiment: scatterplot of 28-day strength vs. fine aggregate....................B-21 

B-25 Factorial experiment: scatterplot of 28-day strength vs. coarse aggregate................B-22 

B-26 Factorial experiment: scatterplot of 28-day strength vs. HRWRA............................B-22 

B-27 Factorial experiment: scatterplot of 28-day strength vs. silica fume .........................B-23 

B-28 Factorial experiment: means plot for 28-day strength ...............................................B-23 

B-29 Factorial experiment: 28-day strength vs. run sequence............................................B-24 

B-30 Factorial experiment: lag plot for 28-day strength ....................................................B-24 

B-31 Factorial experiment: normal probability plot for RCT charge passed .....................B-27 

B-32 Factorial experiment: raw data plot for RCT charge passed......................................B-27 

B-33 Factorial experiment: scatterplot of RCT charge passed vs. w/c...............................B-28 

B-34 Factorial experiment: scatterplot of RCT charge passed vs. fine aggregate..............B-28 

B-35 Factorial experiment: scatterplot of RCT charge passed vs. coarse aggregate..........B-29 

B-36 Factorial experiment: scatterplot of RCT charge passed vs. HRWRA......................B-29 

B-37 Factorial experiment: scatterplot of RCT charge passed vs. silica fume...................B-30 

B-38 Factorial experiment: means plots for RCT charge passed .......................................B-30 

B-39 Factorial experiment: RCT charge passed vs. run sequence......................................B-31 

B-40 Factorial experiment: lag plot for RCT charge passed .............................................B-31 

 C-1 COST homepage...........................................................................................................C-6 

 C-2 “Response Information” form.....................................................................................C-12 

 C-3 First two sections of “Mixture Factors and Information” input form.........................C-14 

 C-4 Third section of “Mixture Factors...” form (mineral admixtures section).................C-15 

 C-5 Third section of “Mixture Factors...” form (chemical admixtures section)...............C-16 

 C-6 Third section of “Mixture Factors...” form (aggregates section)...............................C-17 

 C-7 Fourth section of “Mixture Factors...” form (additional information section) ..........C-18 

 C-8 Portion of an experimental plan generated by COST .................................................C-19 

 C-9 “Run Trial Batches” screen.........................................................................................C-20 
ix

LIST OF FIGURES (continued) 

Figure Page 

C-10 Data entry form for the COST system ........................................................................C-24 

C-11 Analysis menu showing individual analysis tasks......................................................C-25 

C-12 Summary statistics table (output of analysis task 1)...................................................C-26 

C-13 Output of analysis task 2 (counts plot matrix of factors)............................................C-28 

C-14 Output of analysis task 3 (counts plot matrix of factors)............................................C-29 

C-15 Output of analysis task 4.............................................................................................C-30

C-16 Output of analysis task 5A..........................................................................................C-31 

C-17 Output of analysis task 5B ..........................................................................................C-32 

C-18 Output of analysis task 6.............................................................................................C-33

C-19 Output of analysis task 7A..........................................................................................C-34 

C-20 Output of analysis task 7B ..........................................................................................C-35 

C-21 Output of analysis task 8 (for response RCT).............................................................C-36 

C-22 Output of analysis task 9.............................................................................................C-37

C-23 Calculator for predicting responses based on models.................................................C-38 

C-24 Example summary returned by the COST system......................................................C-39 

 
x

LIST OF TABLES 

Table Page 

 1 Number of runs required for CCD experiment for k = 2 to 5 factors)............................10 

 2 Example of ANOVA sequential model sum of squares .................................................11 

 3 Example of ANOVA lack-of-fit test...............................................................................12 

 4 Example of ANOVA model fitting for 1-day strength ...................................................13 

 5 Material volume fraction ranges for mixture experiment...............................................16 

 6 Mixture proportions for mixture experiment ..................................................................17 

 7 Test results and costs (mixture experiment) ...................................................................19 

 8 Sequential model sum of squares for 28-day strength (mixture experiment).................20 

 9 Variable settings corresponding to coded values (factorial experiment)........................27 

 10 Mixture proportions (per m3

) for factorial experiment...................................................29 

 11 Test results and costs (factorial experiment) ..................................................................31 

 12 Sequential model sum of squares for 1-day strength (factorial experiment)..................34 

 13 Lack-of-fit test for 1-day strength (factorial experiment)...............................................34 

 14 ANOVA for 1-day strength model (factorial experiment) .............................................35 

 15 Summary statistics for 1-day strength model (factorial experiment)..............................36 

 16 Summary of analysis tasks and tools in COST...............................................................49 

 A-1 Mixture experiment design in terms of volume fractions of components ................... A-1 

 A-2 Mixture experiment: slump and 1-day strength data .................................................. A-2 

 A-3 Mixture experiment: 28-day strength and RCT charge passed data........................... A-3 

 A-4 Mixture experiment: sequential model of squares for slump...................................... A-4 

 A-5 Mixture experiment: lack-of-fit test for slump ........................................................... A-4 

 A-6 Mixture experiment: model summary statistics for slump ......................................... A-4 

 A-7 Mixture experiment: ANOVA for slump mixture model ........................................... A-5 

 A-8 Mixture experiment: coefficient estimates for slump mixture model......................... A-5 

 A-9 Mixture experiment: adjusted effects for slump mixture model................................. A-5 

 A-10 Mixture experiment: sequential model of sum of squares for 1-day strength ............ A-9 

 A-11 Mixture experiment: lack-of-fit test for 1-day strength .............................................. A-9 

 A-12 Mixture experiment: summary statistics for 1-day strength ....................................... A-9 

 A-13 Mixture experiment: ANOVA for 1-day strength mixture model............................ A-10 

 A-14 Mixture experiment: coefficient estimates for 1-day strength mixture model ......... A-10 

 A-15 Mixture experiment: sequential model of sum of squares for 28-day strength ........ A-15 

 A-16 Mixture experiment: lack-of-fit test for 28-day strength .......................................... A-15 

 A-17 Mixture experiment: model summary statistics for 28-day strength ........................ A-15 

 A-18 Mixture experiment: ANOVA for 28-day strength mixture model.......................... A-16 

 A-19 Mixture experiment: estimated coefficients for 28-day strength mixture model ..... A-16 

 A-20 Mixture experiment: adjusted effects for 28-day strength mixture model ............... A-16 

 A-21 Mixture experiment: sequential model of sum of squares for RCT charge passed .. A-21 

 A-22 Mixture experiment: lack-of-fit test for RCT charge passed.................................... A-21 

 A-23 Mixture experiment: model summary statistics for RCT charge passed .................. A-21 

 A-24 Mixture experiment: ANOVA for RCT charge passed mixture model.................... A-21 

 A-25 Mixture experiment: estimated coefficients for RCT charge passed mixture model A-22 

 A-26 Mixture experiment: adjusted effects for RCT charge passed mixture model ......... A-22 
xi

LIST OF TABLES (continued) 

Table Page 

 B-1 Factorial experiment: design by volume fraction of factors .........................................B-1 

 B-2 Factorial experiment: slump and 1-day strength data..................................................B-2 

 B-3 Factorial experiment: 28-day strength and RCT charge passed data...........................B-3 

 B-4 Factorial experiment: sequential model sum of squares for slump..............................B-4 

 B-5 Factorial experiment: lack-of-fit test for slump...........................................................B-4 

 B-6 Factorial experiment: ANOVA for slump model ........................................................B-4 

 B-7 Factorial experiment: summary statistics for slump model .........................................B-5 

 B-8 Factorial experiment: coefficient estimates for slump model......................................B-5 

 B-9 Factorial experiment: sequential model sum of squares for 1-day strength ..............B-11 

 B-10 Factorial experiment: lack-of-fit test for 1-day strength............................................B-11 

 B-11 Factorial experiment: ANOVA for 1-day strength model .........................................B-11 

 B-12 Factorial experiment: summary statistics for 1-day strength model..........................B-12 

 B-13 Factorial experiment: coefficient estimates for 1-day strength model.......................B-12 

 B-14 Factorial experiment: sequential model sum of squares for 28-day strength ............B-18 

 B-15 Factorial experiment: lack-of-fit test for 28-day strength..........................................B-18 

 B-16 Factorial experiment: ANOVA for 28-day strength model .......................................B-18 

 B-17 Factorial experiment: summary statistics for 28-day strength model........................B-19 

 B-18 Factorial experiment: coefficient estimates for 28-day strength model.....................B-19 

 B-19 Factorial experiment: sequential model sum of squares for RCT charge passed ......B-25 

 B-20 Factorial experiment: lack-of-fit test for RCT charge passed....................................B-25 

 B-21 Factorial experiment: ANOVA for RCT charge passed model.................................B-25 

 B-22 Factorial experiment: summary statistics for RCT charge passed model..................B-26 

 B-23 Factorial experiment: coefficient estimates for RCT charge passed model ..............B-26 

 C-1 Examples of components and responses.......................................................................C-4 

 C-2 Information required for different materials...............................................................C-11 

 C-3 Description of summary statistics provided in analysis task 1 ...................................C-27 

1 

CHAPTER 1 

Introduction 

1.1 Statement of Problem and Project Goals 

The purpose of this project was to investigate the use of statistical experiment design approaches 

in concrete mixture proportioning. These statistical methods are applied in industry to optimize 

products such as gasoline, food products, and detergents. In many cases, the products are, like 

concrete, combinations of several components. Typically, these applications optimize a product 

to meet a number of performance criteria (user-specified constraints) simultaneously, at 

minimum cost. For concrete, these performance criteria could include fresh concrete properties 

such as viscosity, yield stress, setting time, and temperature; mechanical properties such as 

strength, modulus of elasticity, creep, and shrinkage; and durability-related properties such as 

resistance to freezing and thawing, abrasion, or chloride penetration. 

This project was sponsored by the Federal Highway Administration (FHWA) and was performed 

jointly by researchers from FHWA and the National Institute of Standards and Technology 

(NIST) Building Materials and Statistical Engineering Divisions. Both FHWA and NIST hope 

to facilitate the use of high-performance concrete (HPC) in both public and private construction, 

and are currently working to develop tools for optimizing HPC mixture proportions. 

HPC has been referred to as “engineered concrete,” implying that an HPC mixture is not 

specified in a generic recipe, but rather designed to meet project-specific needs [1]. Such a 

definition gives a concrete producer or materials engineer greater than usual latitude in selecting 

constituent materials and defining proportions in an HPC mixture, since fewer or possibly no 

prescriptive constraints, such as minimum cement contents or maximum water-cement (w/c) 

ratios, are included in specifications. HPC mixtures are usually more expensive than 

conventional concrete mixtures because they usually contain more cement, several chemical 

admixtures at higher dosage rates than for conventional concrete, and one or more supplementary 

cementitious materials. As the cost of materials increases, optimizing concrete mixture 

proportions for cost becomes more desirable. Furthermore, as the number of constituent 

materials increases, the problem of identifying optimal mixtures becomes increasingly complex. 

Not only are there more materials to consider, but there also are more potential interactions 

among materials. Combined with several performance criteria, the number of trial batches 

required to find optimal proportions using traditional methods could become prohibitive. 

The general approach to concrete mixture proportioning can be described by the following steps: 

1. Identifying a starting set of mixture proportions. 

2. Performing one or more trial batches, starting with the mixture identified in step 1 above, and 

adjusting the proportions in subsequent trial batches until all criteria are satisfied. 
2 

Current practice in the United States for developing new concrete mixtures often relies upon 

using historical information (i.e., what has worked for the producer in the past) or guidelines for 

mixture proportioning outlined in American Concrete Institute (ACI) 211.1 [2]. Following the 

ACI 211.1 guidelines, an engineer would select and run a first trial batch (selecting proportions 

using ACI 211.1 or historical data), evaluate the results, adjust the proportions of various 

components, and run further trial batches until all specified criteria are met. Typically, this is 

performed by varying one component at a time. While both historical information and ACI 

211.1 can yield a starting point for trial batches, neither method is a comprehensive procedure 

for optimizing mixtures. Historical information may not be valid for materials other than the 

particular ones used in a given project. In ACI 211.1, interactions among the concrete 

constituents cannot be accounted for, and there is no means to achieve an efficiently optimized 

mixture for a given criterion. 

In contrast, statistical experimental design methods are rigorous techniques for both achieving 

desired properties and determining an optimized mixture for a given set of constraints. They are 

used widely in industry to optimize products and processes [3], and have been applied in some 

research studies on improving high-performance concrete [4,5]. They have not, however, been 

applied as a general approach to concrete mixture proportioning. 

Employing statistical methods in the trial batch process does not change the overall approach, 

but it changes the trial batch process. Rather than selecting one starting point, a set of trial 

batches covering a chosen range of proportions for each component is defined according to 

established statistical procedures [3]. Trial batches are then carried out, test specimens are 

fabricated and tested, and results are analyzed using standard statistical methods. These methods 

include fitting empirical models to the data for each performance criterion. In these models, each 

response (resultant concrete property) such as strength, slump, or cost, is expressed as an 

algebraic function of factors (individual component proportions) such as w/c, cement content, 

chemical admixture dosage, and percent pozzolan replacement. 

After a response can be characterized by an equation (model), several analyses are possible. For 

instance, a user could determine which mixture proportions would yield one or more desired 

properties. A user also could optimize any property subject to constraints on other properties. 

Simultaneous optimization to meet several constraints is also possible. For example, one could 

determine the lowest cost mixture with strength greater than a specified value, air content within 

a given range, and slump within a given range. A method for optimizing several responses 

simultaneously is described later in the report. 

Mechanistic (or semimechanistic) models that were developed from results of fundamental and 

applied materials research have also been used as a basis for mixture proportioning methods [6]. 

An advantage of this approach is that it does not require trial batches to obtain the models; 

however, some trial batches most likely would be needed to adjust proportions because of 

differences in material properties at the local level. It is unlikely that a mechanistic model would 

be able to account for all possible differences in local materials. The advantage of the trial batch 

approach is that the project-specific materials are used and accounted for in the model. 
3 

An additional advantage of the statistical approach is that the expected properties (responses) can 

be characterized by an uncertainty (variability). This has important implications for 

specifications and for production. When an empirical model equation is used to determine 

mixture proportions that yield a desired strength, the model equation gives only the expected 

mean strength; that is, if replicate mixtures were made, the model equation would predict the 

mean value. This is not an appropriate target value for specifications, because in the long run, 

the strength would be below that value half of the time. Instead, to ensure that most of the 

strength test results would comply with specifications, a producer would select target values for 

the mean strength to account for the variability and to ensure that, for example, 95 percent of the 

results would be expected to meet or exceed the specified value. 

A disadvantage of the statistical approach is that it requires an initial investment of time and 

money for planning and performing trial batches and tests. Additionally, knowledge of good 

experimentation procedures and some knowledge of statistical analysis is needed. Statistical 

computer programs are available to perform both experiment design and analysis, but knowing 

how to interpret and ensure the validity of statistical models is important. For this reason, the 

second objective of this project was to develop an interactive Web site to provide users with 

rudimentary knowledge and lead them step by step through a mixture proportioning process 

using statistical methods. The aim was not to provide a comprehensive, user-friendly software 

package, but rather to introduce producers and engineers to these methods and to provide 

sufficient results and guidance on interpretation to allow them to see potential advantages of the 

approach. 

Although these methods require a commitment of time and money upfront, they have the 

potential to save money during construction. Reducing the concrete material cost by $20 per 

cubic meter (m3

) could result in savings of $40,000 per km of 30-cm thick, two-lane concrete 

pavement. 

1.2 Scope of Report 

The report is organized as follows: Chapter 1 introduces the problem and the project goals, and 

describes the scope. Chapter 2 provides background on the statistical concepts used in this 

project, including response surface methodology (RSM) and its components: experiment design, 

model fitting and validation, and optimization. Chapter 3 describes the laboratory experiment 

using a mixture experiment design approach, and chapter 4 describes a laboratory experiment 

using a mathematically independent variable (MIV), or factorial, approach. Chapter 5 describes 

the development of the interactive Web site, the Concrete Optimization Software Tool (COST). 

References are provided after chapter 5. Appendices A (mixture experiment) and B (factorial 

experiment) contain experiment designs, test data, data analysis and model fitting (tables and 

graphs) from the laboratory experiments. Appendix C contains the COST User’s Guide, which 

describes the COST system and its use in detail.

5

CHAPTER 2 

Background on Statistical Methods 

2.1 Response Surface Methodology 

Response surface methodology (RSM) consists of a set of statistical methods that can be used to 

develop improve, or optimize products [3]. RSM typically is used in situations where several 

factors (in the case of concrete, the proportions of individual component materials) influence one 

or more performance characteristics, or responses (the fresh and hardened properties of the 

concrete). RSM may be used to optimize one or more responses (e.g., maximize strength, 

minimize chloride penetration), or to meet a given set of specifications (e.g., a minimum strength 

specification or an allowable range of slump values). There are three general steps that comprise 

RSM: experiment design, modeling, and optimization. Each of these is described below. 

Concrete is a mixture of several components. Water, portland cement, and fine and coarse 

aggregates form a basic concrete mixture. Various chemical and mineral admixtures, as well as 

other materials such as fibers, also may be added. For a given set of materials, the proportions 

of the components directly influence the properties of the concrete mixture, both fresh and 

hardened. 

2.2 Experiment Design 

Consider a concrete mixture consisting of q component materials (where q is the number of 

component materials). Two experiment design approaches can be applied to concrete mixture 

optimization: the classic mixture approach, in which the q mixture components are the variables, 

[7] and the mathematically independent variable (MIV) approach, in which q mixture 

components are transformed into q-1 independent mixture-related variables [8]. Each technique 

has advantages and disadvantages. In the classic mixture approach, the sum of the proportions 

must be 1; therefore the variables are not all independent. This allows the experimental region of 

interest to be defined more naturally, but the analysis of such experiments is more complicated. 

The MIV approach, with the variables independent, permits the use of classical factorial and 

response surface designs [9], but has the undesirable feature that the experimental region changes 

depending on how the q mixture components are reduced to q-1 independent factors. 

In this report, the MIV approach is referred to as the factorial approach because factorial 

experiment designs form the basis of the approach. The following sections present general 

(nonrigorous) descriptions of each method (for a detailed discussion of these methods, see 

reference 3). 

2.2.1 Mixture Approach 

In the mixture approach, the total amount (mass or volume) of the product is fixed, and the 

settings of each of the q components are proportions. Because the total amount is constrained to 

sum to one, only q-1 of the factors (component variables) can be chosen independently. 
6

As a simple (hypothetical) example of a mixture experiment, consider concrete as a mixture of 

three components: water (x1), cement (x2), and aggregate (x3), where each xi represents the 

volume fraction of a component. Assume the coarse-to-fine aggregate ratio is held constant. 

The volume fractions of the components sum to one, and the region defined by this constraint is 

the regular triangle (or simplex) shown in figure 1. The axis for each component xi extends from 

the vertex it labels (xi = 1) to the midpoint of the opposite side of the triangle (xi = 0). The 

vertex represents the pure component. For example, the vertex labeled x1 is the pure water 

mixture with x1 = 1, x2 = 0, and x3 = 0, or (1,0,0). The point where the three axes intersect, with 

coordinates (1/3,1/3,1/3), is called the centroid. 

A good experiment design for studying properties over the entire region of a three-component 

mixture would be the simplex-centroid design shown in figure 2 (this example is included as an 

illustration only, since much of this region would not represent either feasible or workable 

x1 (1,0,0)

x3 x (0,0,1) 2 (0,1,0)

x1 (1,0,0)

x3 x (0,0,1) 2 (0,1,0)

Figure 2. Simplex-centroid design for three variables 

Figure 1. Example of triangular simplex region from three-component 

mixture experiment 

x3

=

0

x1 = 0

x2

=

0

x1 (1,0,0)

x3 (0,0,1) x2 (0,1,0)

x3

=

0

x1 = 0

x2

=

0

x1 (1,0,0)

x3 (0,0,1) x2 (0,1,0)
7

concrete mixtures). The points shown in figure 2 represent mixtures included in the experiment 

and include all vertices, midpoints of edges, and the overall centroid. 

All responses (properties) of interest would be measured for each mixture in the design and 

modeled as a function of the components. Typically, polynomial functions are used for 

modeling, but other functional forms can also be used. For three components, the linear 

polynomial model for a response y is: 

where the bi * are constants and e, the random error term, represents the combined effects of all 

variables not included in the model. For a mixture experiment, 1 x1 + x2 + x3 = ; therefore, the 

model can be reparameterized in the form: 

using ( ) 1 2 3

*

0

*

0 b = b ⋅ x + x + x . This form is called the Scheffé linear mixture polynomial [7]. 

Similarly, the quadratic polynomial: 

can be reparameterized as: 

using ) 1 (1 2 3

2

1 x = x ⋅ − x − x , ) 2 (1 1 3

2

2 x = x ⋅ − x − x , and ) 3 (1 1 2

2

3 x = x ⋅ − x − x . 

Since feasible concrete mixtures do not exist over the entire region shown in figures 1 and 2, a 

subregion of the full simplex containing the range of feasible mixtures must be defined by 

constraining the component proportions. An example of a possible subregion for the threecomponent example is shown in figure 3. It is defined by the following volume fraction 

constraints (where x1 = water, x2 = cement, x3 = aggregate): 

0.15 ≤ x1 ≤ 0.25 

0.10 ≤ x2 ≤ 0.20 

0.60 ≤ x3 ≤ 0.70 

In this case the simplex designs are usually no longer appropriate and other designs are used [3]. 

y = b x +b x +b x +e 1 1 2 2 3 3

y = b + b x + b x + b x3 + e * 2 3 * 1 2 *

1 *0 (1)

(2)

y = b +b x +b x +b x + b x x + b x x + b x x + b x + b x + b x + e

2

3

*

33

2

2

*

22

2

1

*

2 3 11

*

1 3 23

*

1 2 13

*

3 12

*

2 3

*

1 2

*

0 1 (4)

y = b x +b x +b x + b x x + b x x + b x x + e 1 1 2 2 3 3 12 1 2 13 1 3 23 2 3

(3)
8

The advantage of the mixture approach is that the experimental region of interest is defined more 

naturally; however, analysis of the results can be complicated, especially if the number of 

components is greater than three. 

2.2.2 Factorial (MIV) Approach 

In the factorial approach, the q components of a mixture are reduced to q-1 independent variables 

using the ratio of two components as an independent variable. In the case of concrete, w/c is a 

natural choice for this ratio variable. For the situation with q-1 independent variables, a 2q-1

factorial design forms the backbone of the experiment. This design consists of several factors 

(variables) set at two different levels. As a simple example, consider a concrete mixture 

composed of four components: water, cement, fine aggregate, and coarse aggregate. Three 

independent factors, or variables, xk , that can be selected to describe this system are x1 = w/c (by 

mass), x2 = fine aggregate volume fraction, and x3 = coarse aggregate volume fraction. 

Reasonable ranges for these variables might be: 

 0.40 ≤ x1 ≤0.50 

 0.25 ≤ x2 ≤0.30 

 0.40 ≤ x3 ≤0.45 

The levels for this example would be 0.40 and 0.50 for x1, 0.25 and 0.30 for x2, and 0.40 and 0.45 

for x3. To simplify calculations and analysis, the actual variable ranges are usually transformed 

to dimensionless coded variables with a range of ±1. In this example, the actual range of 

0.40 ≤ x1 ≤ 0.50 would translate to a coded range of -1 ≤ x1 ≤1. Intermediate values of x1 would 

translate similarly (e.g., the actual value of 0.45 would translate to a coded value of zero). The 

general equation used to translate from coded to uncoded is as follows: 

( ) 2

( 1)

min max min x x

x

x x coded

actual ⋅ − + = + (5)

x1 (1,0,0)

x3 (0,0,1) x2 (0,1,0)

x1 = .15

x2 = .20

x1 = .25

x2 = .10

x3 = .60

x3 = .70

x1 (1,0,0)

x3 (0,0,1) x2 (0,1,0)

x1 (1,0,0)

x3 (0,0,1) x2 (0,1,0)

x1 = .15

x2 = .20

x1 = .25

x2 = .10

x3 = .60

x3 = .70

Figure 3. Example of subregion of full simplex containing range of feasible mixtures 
9

where xactual is the uncoded value, xmin and xmax are the uncoded minimum and maximum values 

(corresponding to –1 and +1 coded values), and xcoded is the coded value to be translated.

Suppose that the specifications for this mixture require a slump of 75 to 150 mm and a 28-day 

strength of 30 MPa. These specified properties are called the responses, or dependent variables, 

yi, which are the performance criteria for optimizing the mixture. For concrete, the responses 

may be any measurable plastic or hardened properties of the mixture. Cost may also be a 

response. 

As with the mixture approach, empirical models are fit to the data, and polynomial models 

(linear or quadratic) typically are used. Equation 6 illustrates the general case of the full 

quadratic model for k =3 independent variables: 

In equation 6, the ten coefficients are represented by the bk and e is a random error term 

representing the combined effects of variables not included in the model. The interaction terms 

(xixj) and the quadratic terms (xi

2

) account for curvature in the response surface. 

The central composite design (CCD), an augmented factorial design, is commonly used in 

product optimization. A complete CCD experiment design allows estimation of a full quadratic 

model for each response. A schematic layout of a CCD for k = 3 independent variables is shown 

in figure 4. The design consists of 2k (in this case, 8) factorial points (filled circles in figure 4) 

representing all combinations of coded values xk = ±1, 2*k (in this case, 6) axial points (hollow 

circles in figure 4) at a distance ±α from the origin, and at least 3 center points (hatched circle in 

figure 4) with coded values of zero for each xk . The value of α usually is chosen to make the 

design rotatable1

, but there are sometimes valid reasons to select other values [3]. 

 

 

1

If a design is rotatable, predicted values should have equal variances at locations equidistant from the origin. 

y = b0 1 1 2 2 3 3 12 1 2 13 1 3 23 2 3 11 +b x +b x +b x +b xx +b xx +b xx +b x +b x +b x + e 2

1 22

2

2 33

2

3

Figure 4. Schematic of a central composite design for three factors 

(6)
10

In the absence of curvature, a model with only linear terms would be sufficient, and the factorial 

portion of the CCD is a valid design by itself in that case. However, the presence or absence of 

significant curvature often is not known with certainty at the start. An advantage of the CCD 

over the mixture approach is that the CCD can be run sequentially in two blocks. The first block 

would consist of the factorial points (all combinations of xi = ±1) and some center points (at least 

3), and the second block would consist of the axial points (points along each axis at distance α

from the origin) and additional center points (at least 2). This approach allows analysis of the 

factorial portion before the axial portion is run. If curvature is insignificant based on the 

factorial portion, the additional runs are not necessary. 

As shown in table 1, the number of coefficients in the quadratic model increases with k, and the 

number of trial batches required using a CCD begins to increase significantly for k > 5. 

Table 1. Number of runs required for CCD experiment for k = 2 to 5 factors 

k Factorial Axial Center* Total 

2 4 4 5 13 

3 8 6 5 19 

4 16 8 5 29 

5 16** 10 5 31 

*assumes 3 CP for factorial portion and 2 CP for axial portion 

**for k=5, a half-fraction of the full factorial is sufficient to estimate all linear terms 

and 2-factor interactions without confounding. Thus, 25-1 or 16 factorial points are 

usually sufficient

Therefore, using a CCD to optimize a concrete mixture of more than six components may be 

uneconomical. In such cases, one could identify the most important factors and limit them to 

five or fewer. For example, if the cementitious materials and chemical admixtures were the most 

important components, they would be varied, while the amounts of coarse and fine aggregate 

would be held constant. 

Laboratory experiments were conducted using the mixture and factorial approaches to see if one 

was more appropriate for concrete mixture optimization. The experiments are described in 

chapters 3 and 4. The adaptability of each method for use as an interactive, Web-based program 

was also considered in developing the COST software, described in chapter 5. 

2.3 Model Fitting and Validation 

The polynomial models described in sections 2.2.1 and 2.2.2 are fit to data using analysis of 

variance (ANOVA) and least squares techniques [9]. Many statistical software packages have 

the capability to perform these analyses and fits. Once a model has been fit, it is important to 

verify the adequacy of the chosen model quantitatively and graphically. 

Although the models (polynomials) are slightly different for the classical mixture approach and 
11

the factorial approach, many of the steps involved in model selection and fitting are the same. 

The first step in each case is to perform ANOVA to select the appropriate type of model (linear, 

quadratic, etc.). Sequential F-tests are performed, starting with a linear model and adding terms 

(quadratic, and higher if appropriate). Under the “Source” column of the ANOVA table, the line 

labeled “Linear” indicates the significance of adding linear terms, and the line labeled 

“Quadratic” indicates the significance of adding quadratic terms. The column labeled “DF” 

shows the degrees of freedom for each source. The F-statistic is calculated for each type of 

model, and the highest order model with significant terms normally would be chosen. 

Significance is judged by determining if the probability that the F-statistic calculated from the 

data exceeds a theoretical value. The probability decreases as the value of the F-statistic 

increases. If this probability is less than 0.05 (typically, although other levels of significance 

could be used), the terms are significant and their inclusion improves the model. An example of 

an ANOVA table for sequential model sum of squares is shown in table 2. 

Table 2. Example of ANOVA sequential model sum of squares 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 329.27 1 329.27 – – 

Linear 96.94 5 19.39 45.64 < 0.0001 

Quadratic 6.36 15 0.42 1.00 0.5017 

Special Cubic (aliased) 3.35 7 0.48 1.26 0.3727 

Cubic (aliased) 0.00 0 – – – 

Residual 3.03 8 0.38 – – 

Total 438.95 36 12.19 – – 

In this example, the linear model is the highest order model with significant terms (Prob > F is 

less than 0.05); therefore, it would be the recommended model for this data. Typically, the 

selected model will be the highest order polynomial where additional terms are significant and 

the model is not aliased. 

Once the type of model (e.g., linear, quadratic, etc.) is selected, the second step is to perform a 

lack-of-fit test, also using ANOVA, to compare the residual error to the pure error from 

replication. Table 3 is an example of ANOVA for lack of fit. If residual error significantly 

exceeds pure error, the model will show significant lack of fit, and another model may be more 

appropriate. 

The desired result in a lack-of-fit test is that the model selected in step 1 will not show significant 

lack of fit (i.e., the F test will be insignificant). If the “Prob > F” value is less than the desired 

significance level (often .05), this indicates significant lack of fit. 

Several summary statistics can be calculated for a model and used to compare models or verify 

model adequacy. These statistics include root mean square error (RMSE), adjusted r2

, 
12

Table 3. Example of ANOVA lack-of-fit test 

 

predicted r2

, and prediction error sum of squares (PRESS). The RMSE is the square root of the 

mean square error, and is considered to be the standard deviation associated with experimental 

error. The adjusted r2

 is a measure of the amount of variation about the mean explained by the 

model, adjusted for the number of parameters in the model2

. The predicted r2 measures the 

amount of variation in new data explained by the model. PRESS measures how well the model 

fits each point in the design. To calculate PRESS, the model is used to estimate each point using 

all of the design points except the one being estimated. PRESS is the sum of the squared 

differences between the estimated values and the actual values over all the points. A good model 

will have a low RMSE, a large predicted r2

, and a low PRESS. 

After a model is selected, standard linear regression techniques (least squares) are used to fit the 

model to the data. ANOVA is performed and an overall F-test and lack-of-fit test confirm the 

applicability of the model. Summary statistics (r2

, adjusted r2, PRESS, etc.) and the standard 

error for each model coefficient also are calculated. 

For the factorial approach, an iterative model fitting process was used in this research. First, a 

full quadratic (or linear, if applicable) model is assumed, and significance tests (t-tests) are 

performed on each model coefficient. Insignificant terms are removed and the fitting process is 

repeated using a partial quadratic (or linear, if applicable) model. The significance tests are 

repeated and insignificant terms, if any, are removed. The process is repeated until there are no 

insignificant terms. At this point, if the model contains two-factor interaction or quadratic terms, 

it is checked for hierarchy. Hierarchical terms are linear terms that may be insignificant by 

themselves but are part of significant higher order terms. For example, x1 and x3 are hierarchical 

terms of x1x3, a two-factor interaction term. If x1x3 is a significant term in the model, x1 and x3

are usually included in the model to maintain hierarchy. A hierarchical model allows for 

conversion of models between different sets of units (for a model involving temperature, 

conversion from F to C, for example). 

Table 4 shows an ANOVA table for a selected model from a factorial experiment. Using the 

iterative approach, a reduced quadratic model was fit to the data. Note that terms B, C, and E are 

not significant (“Prob > F” >.05) but were added back into the model to make it hierarchical. 

 

2

The adjusted r2 differs from the “standard” r2, which is not adjusted for the number of parameters. The standard r2

can be made larger by adding more parameters to the model. This does not necessarily mean the model with more 

parameters is a better model. 

Source Sum of Squares DF Mean Square F Value Prob > F 

Linear 6271.93 22 285.09 1.17 0.4335 

Quadratic 2164.86 7 309.27 1.27 0.3703 

Special Cubic (aliased) 0.00 0 – – – 

Cubic (aliased) 0.00 0 – – – 

Pure Error 1950.91 8 243.86 – – 
13

Table 4. Example of ANOVA model fitting for 1-day strength 

Source Sum of 

Squares DF Mean Square F Value Prob > F 

Model 240.87 8 30.11 27.76 < 0.0001 

 A 213.26 1 213.26 196.66 < 0.0001 

 B 0.48 1 0.48 0.45 0.5113 

 C 0.04 1 0.04 0.04 0.8433 

 E 2.06 1 2.06 1.90 0.1819 

 A2 6.20 1 6.20 5.72 0.0257 

 AC 5.15 1 5.15 4.75 0.0404 

 AE 7.16 1 7.16 6.60 0.0175 

 BC 6.51 1 6.51 6.00 0.0227 

Residual 23.86 22 1.08 – – 

 Lack of fit 19.08 18 1.06 0.89 0.6248 

 Pure error 4.78 4 1.19 – – 

Corr. total 264.72 30 

Once the model fitting is performed, the next step is residual analysis to validate the assumptions 

used in the ANOVA. This analysis includes calculating case statistics to identify outliers and 

examining diagnostic plots such as normal probability plots and residual plots. If these analyses 

are satisfactory, the model is considered adequate, and response surface (contour) plots can be 

generated. Contour plots can be used for interpretation and optimization. 

2.4 Optimization 

When appropriate models have been established, several responses can be optimized 

simultaneously. Optimization may be performed using mathematical (numerical) or graphical 

(contour plot) approaches. Generally, graphical optimization is limited to cases in which there 

are only a few responses. 

Numerical optimization requires defining an objective function (called a desirability or score 

function) that reflects the levels of each response in terms of minimum (zero) to maximum (one) 

desirability. One approach uses the geometric mean of the desirability functions for each 

individual response, where n is the number of responses to be optimized [10]: 

n

n D d d d 1

( ) = 1 × 2 ×K× (7)
14

Another approach is to use a weighted average of desirability functions: 

where n is the number of responses and wi are weighting functions ranging from 0 to 1. 

Several types of desirability functions can be defined. Common types of desirability functions 

are shown in figure 5. 

These functions can also be expressed mathematically as well. For example, a linear desirability 

function where minimum is best would be expressed as: 

for the range Ai ≤ Yi ≤ Bi and with wi = 1. 

Once the desirability functions (and weights, if used) are defined for each response, optimization 

may proceed. 

As an alternative to rigorous numerical methods, desirability can be evaluated by superimposing 

a grid of points at equal spacing over the experimental region and evaluating desirability at each 

point. The point(s) of maximum desirability can be found by sorting the results or by creating 

contour plots of desirability over the grid area. 

n

w d w d w d

D n n ( ) 1 × 1 + 2 × 2 + + × = K

minimum

linear (decr.) within range

maximum target

linear (incr.)

minimum

linear (decr.) within range

maximum target

linear (incr.)

Figure 5. Examples of desirability functions

wi

i i

i i

i B A

B Y d ⎥

⎦

⎤ ⎢

⎣

⎡

−

− =

(8)

(9)
15

CHAPTER 3 

Laboratory Experiment Using Mixture Approach 

3.1 Introduction 

This chapter describes the application of a statistically designed mixture experiment to the 

problem of optimizing properties of HPC. In a mixture experiment, the total amount (mass or 

volume) of the mixture is fixed, and the factors or component settings are proportions of the total 

amount. For concrete, the sum of the volume fractions is constrained to sum to one (as in the 

ACI mix design approach). Since the volume fractions must sum to unity, the component 

variables in a mixture experiment are not independent. 

A mixture experiment was designed to find the optimum proportions for a concrete mix meeting 

the following conditions: slump of 50 to 100 mm, 1-day compressive strength of 22.06 MPa, 28-

day compressive strength of 51.02 MPa, 42-day charge passed in American Society for Testing 

and Materials (ASTM) C1202 “rapid chloride” test (RCT) less than 700 coulombs, and minimum 

cost. The materials (components) used included water, cement, silica fume, high-range waterreducing admixture (HRWRA), coarse aggregate, and fine aggregate. 

3.2 Selection of Materials, Proportions, and Constraints 

The proportions for the six-component mixture experiment initially were selected in terms of 

volume fraction and converted to weights for batching. The minimum and maximum levels of 

each component were chosen based on typical volume fractions for non-air-entrained concrete

with the constraint that the volume fractions sum to unity. In addition to the individual 

constraints on each component, the paste fraction of the concrete (water, cement, silica fume, and 

HRWRA) was required to range from 25 to 35 percent by volume. Although air is incorporated 

into concrete during mixing, it is not an initial component and therefore was not considered to be 

a component of the mixture. Ignoring the air content as a mix component affects yield 

calculations, but these are not important for the small trial batches and can be adjusted later after 

a final mix is selected. 

The materials used in this study included a Type I/II portland cement, tap water, #57 crushed 

limestone coarse aggregate, natural sand, silica fume (in slurry form), and a naphthalene 

sulfonate-based HRWRA meeting ASTM C494 Type F/G. The final volume fraction ranges of 

the 6 mixture components are shown in table 5. The volume fractions were converted to 

corresponding weights using the specific gravities and percent solids (where applicable) obtained 

from laboratory testing or from the material supplier. 

3.3 Experiment Design Details 

Selecting an appropriate experiment design depends on several criteria, such as ability to 

estimate the underlying model, ability to provide an estimate of repeatability, and ability to check 

the adequacy of the fitted model. The “best” experiment design depends on the choice of an 
16

Table 5. Material volume fraction ranges for mixture experiment 

underlying model which will adequately explain the data. For this experiment, the following 

quadratic Scheffé polynomial was chosen as a reasonable model for each property as a function 

of the six components: 

This model is an extension of equation 4 for the 6-component case. Because there are 21 

coefficients in the model, the design must have at least 21 runs (21 distinct mixes) to estimate 

these coefficients. In addition to the 21 required runs, 7 additional runs (distinct mixes) were 

included to check the adequacy of the fitted model, and 5 mixes were replicated to provide an 

estimate of repeatability. The replicates were required to test the statistical significance of the 

fitted coefficients. Finally, a single mix was replicated during each week of the experiment to 

check statistical control of the fabrication and measurement process. In all, a total of 36 mixes 

were planned. 

Commercially available computer software for experiment design was used to design and 

analyze the experiment. The program selected 36 points from a list of candidate points that is 

known to include the best points for fitting a quadratic polynomial. A modified-distance design 

was chosen to ensure that the design selected could estimate the quadratic mixture model while 

spreading points as far away as possible from one another. 

Table 6 summarizes the mixture proportions used in the experiment. The run order was 

randomized to reduce the effects of extraneous variables not explicitly included in the 

experiment. The first three mixes were repeated at the end of the program (runs 37, 38, and 39), 

because an incorrect amount of water was used in batching the mixes. The test results from the 

incorrectly batched mixes were not included in the subsequent analysis. Of the final 36 mixes, 8 

were replicates (one each from mixes 5, 11, 20, 38, 71 and three from mix 127). 

3.4 Specimen Fabrication and Testing

Thirty-nine batches of concrete, each approximately .04 m3

 in volume, were prepared over a 

four-week period. A rotating-drum mixer with a 0.17 m3

 capacity was used to mix the concrete. 

Component ID Minimum 

volume fraction 

Maximum 

volume fraction 

 Water x1 .16 .185 

 Cement x2 .13 .15 

 Silica fume x3 .013 .027 

 HRWRA x4 .0046 .0074 

 Coarse aggregate x5 .40 .4424 

 Fine aggregate x6 .25 .2924 

y = b x + ... + b x + b x x + ... + b x x + e 1 1 6 6 12 1 2 56 5 6 (10)
17

Table 6. Mixture proportions for mixture experiment 

Design 

ID 

Run Order w/(c+sf)

 

Water 

(kg) 

Cement

(kg) 

Silica 

Fume 

(kg) 

HRWRA

(l) 

Coarse 

Aggregate

(kg) 

Fine 

Aggregate 

(kg) 

 5(r) 7, 22 122.3 312.9 45.4 3.52 867.6 506.3 0.35 

 11(r) 6, 23 141.4 312.9 21.9 3.52 845.3 506.3 0.43 

 13 15 122.3 312.9 21.9 3.52 810.1 592.2 0.37 

 15 2*, 38 126.6 361.1 45.4 5.66 810.1 506.3 0.32 

 16 8 122.3 312.9 21.9 3.52 895.9 506.3 0.37 

 20(r) 13, 34 141.4 312.9 21.9 3.52 810.1 541.8 0.43 

 22 4 141.4 354.8 21.9 3.52 810.1 506.3 0.38 

 28 16 122.3 312.9 45.4 3.52 810.1 563.8 0.35 

 37 30 122.3 337.0 45.4 5.66 810.1 537.9 0.33 

 38(r) 3*, 26, 39 135.0 341.1 45.4 3.52 810.1 506.3 0.36 

 48 28 131.8 312.9 21.9 5.66 810.1 561.2 0.41 

 63 27 131.8 312.9 45.4 5.66 836.6 506.3 0.38 

 65 31 122.3 337.0 45.4 5.66 841.7 506.3 0.33 

 66 25 122.3 312.9 45.4 5.66 836.0 532.2 0.35 

 70 29 122.3 361.1 21.9 4.59 810.1 548.8 0.33 

 71(r) 5, 35 122.3 361.1 21.9 5.66 829.9 526.1 0.33 

 78 11 141.4 312.9 45.4 5.66 810.7 506.9 0.41 

 87 24 122.3 312.9 21.9 3.52 853.0 549.2 0.37 

 89 19 122.3 337.0 21.9 3.52 810.1 571.9 0.35 

 91 9 141.4 312.9 21.9 5.66 824.9 521.1 0.43 

 98 17 122.3 337.0 21.9 3.52 875.7 506.3 0.35 

 101 10 130.8 361.1 21.9 3.52 832.8 506.3 0.35 

 103 14 122.3 361.1 21.9 4.59 852.6 506.3 0.33 

 110 21 130.8 361.1 21.9 3.52 810.1 529.0 0.35 

 116 33 131.8 312.9 45.4 5.66 810.1 532.8 0.38 

 123 36 122.3 337.0 33.6 4.59 834.4 530.6 0.34 

 127(c) 1*, 12, 18, 

32, 37 131.5 335.8 21.9 4.59 829.9 526.1 0.38 

 163 20 126.6 323.3 27.8 5.12 857.5 513.6 0.37 

Notes: Aggregate masses are in dry condition 

 (r) replicated mix 

 (c) control mix 

 * mix that was repeated due to incorrect batching 
18

Each batch included sufficient concrete for two slump tests, two fresh air content (ASTM C231) 

tests, two unit weight tests, and ten 100 mm x 200 mm cylinders. The cylinders were fabricated 

in accordance with ASTM C192. To obtain adequate consolidation, cylinders for concretes with 

slumps less than 50 mm were vibrated on a vibrating table; otherwise, the cylinders were rodded. 

The cylinders were covered with plastic and left in the molds for 22 hours, after which they were 

stripped and placed in limewater-filled curing tanks for moist curing at 23 ±2°C. 

Compressive strength tests (ASTM C39) were conducted on the cylinders at the ages of 1 day 

and 28 days. In most cases, three cylinders were tested for each age. A fourth test was 

performed in some cases if one result was significantly lower or higher than the others. Before 

testing, the cylinder ends were ground parallel to meet the ASTM C39 requirements using an 

end-grinding machine designed for this purpose. The three remaining cylinders from each batch 

were used for “rapid chloride” testing according to ASTM C1202. Three 50-mm thick slices 

taken from the middle sections of the concrete cylinders were tested at an age of 42 days. 

3.5 Results and Analysis 

3.5.1 Measured Responses 

The average values for slump, 1-day strength, 28-day strength, and RCT charge passed for each 

batch are shown in table 7, along with the estimated cost per cubic meter of concrete. The cost 

of each batch was calculated from the mix proportions using approximate costs for each 

component material obtained from a local ready-mix concrete producer. For each of the four 

responses, a model was fit using least-squares methods, validated (by examining the residuals for 

trends and outliers), and interpreted graphically using contour and trace plots. The statistical 

analysis is described in detail for 28-day strength. The analyses for the other properties were 

performed in a similar manner. ANOVA tables, model statistics, and plots for all responses are 

included in Appendix A. 

3.5.2 Model Identification and Validation for 28-Day Strength 

In this section, a detailed description of the process of model identification and validation is 

provided for the response 28-day strength. The models for other responses were identified and 

validated in the same way. Those models are presented in the next section. 

The first step in the analysis is to select a plausible model. Even though the experiment design 

used permits estimation of a quadratic model, a linear model may provide a better fit to the data. 

ANOVA is used to assess the different models. 

ANOVA results for 28-day strength are shown in table 8. In this table, each row tests whether 

the coefficients of certain model terms are equal to zero. For example, the row with source 

“Linear” tests whether the coefficients of the linear terms are equal to zero. A low value (say, 

less than 0.05) of “Prob > F” , also called the p-value, supports the conclusion that the 

coefficients differ from zero and should be included in the model. For the data in table 7, the 

“Prob > F” is 0.0011; therefore, the linear terms should be included in the model. The row with
19

Table 7. Test results and costs (mixture experiment)

Design 

ID Run Slump (mm) 

1-Day Str

(MPa) 

28-Day 

Str 

(MPa) 

42-Day 

RCT 

(coulombs) 

Cost 

($/ m3) 

 22 4 67 21.5 48.2 1278 95.18 

 71 5 57 27.0 55.2 862 102.22 

 11 6 102 16.8 48.5 1162 91.32 

 5 7 13 22.4 48.5 387 118.85 

 16 8 35 21.6 53.1 776 92.20 

 91 9 200 16.8 60.4 1027 96.89 

 101 10 22 26.6 53.6 744 96.24 

 78 11 127 19.2 51.7 492 123.56 

 127 12 99 21.5 50.2 842 96.67 

 20 13 118 18.2 50.9 903 91.32 

 103 14 64 27.4 54.6 583 99.42 

 13 15 57 21.8 53.2 684 92.20 

 28 16 29 22.2 53.6 292 118.85 

 98 17 32 25.3 51.9 604 94.41 

 127 18 92 22.3 54.1 847 96.67 

 89 19 38 21.8 54.3 720 94.41 

 163 20 95 22.1 60.8 554 103.80 

 110 21 51 24.7 53.2 792 96.24 

 5 22 25 23.4 54.1 348 118.85 

 11 23 114 16.5 48.0 968 91.32 

 87 24 67 22.9 51.0 700 92.20 

 66 25 76 24.7 59.8 316 124.44 

 38 26 29 23.0 53.2 390 120.85 

 63 27 124 21.7 55.2 302 123.99 

 48 28 171 23.0 58.1 682 97.34 

 70 29 51 27.5 54.5 505 99.42 

 37 30 35 27.3 56.0 245 126.65 

 65 31 32 27.2 51.1 310 126.65 

 127 32 121 22.4 57.2 636 96.67 

 116 33 114 23.9 56.2 356 123.99 

 20 34 127 18.6 51.6 820 91.32 

 71 35 108 28.8 65.3 553 102.22 

 123 36 99 26.6 61.0 340 110.53 

 127 37 102 24.2 54.6 640 96.67 

 15 38 51 28.8 58.1 239 128.68 

 38 39 25 23.6 54.5 332 120.85
20

Table 8. Sequential model sum of squares for 28-day strength (mixture experiment)

Source Sum of Squares DF Mean Square F Value Prob > F 

 Mean 106212.98 1 106212.98 – – 

 Linear 257.52 5 51.50 5.46 0.0011 

 Quadratic 135.19 15 9.01 0.92 0.5665 

 Residual 147.62 15 9.84 – – 

 Total 106753.31 36 2965.37 – – 

 source “Quadratic” tests whether any quadratic coefficients differ from zero. Since the “Prob 

>F” of 0.5667 exceeds 0.05, the quadratic terms should not be included in the model. 

A lack-of-fit test can also be performed using ANOVA. To do so, the residual sum of squares is 

partitioned into lack-of-fit and pure error (from replicates) components. The mean squares and 

F statistic are calculated, and the “Prob > F” is determined. The desired result is an insignificant 

lack-of-fit, indicated by a “Prob > F” greater than 0.05. For 28-day strength, the lack-of-fit test 

for the linear model gives “Prob > F” equal to 0.7193, which is insignificant (the ANOVA tables 

for the lack-of-fit test for 28-day strength and the other responses are provided in Appendix A). 

The resulting linear model for 28-day strength (y1), fit by least squares, is 

with the residual standard deviation s = 3.07 MPa. The residual standard deviation s is defined 

as: 

where the number of observations n = 36 and the number of parameters in the fitted model p = 6. 

A value of s near the repeatability value (replicate standard deviation) is an indication of an 

adequately fitting model. The repeatability value is 3.39 MPa, which is close to s. 

Residual plots are used to validate the fitted model. The residuals are the deviations of the 

observed data from the fitted values, yi - y^i. The residual yi - y^i. estimates the error terms ei in the 

model. The ei are assumed to be random and normally distributed with mean zero and constant 

standard deviation. The residuals, which estimate these errors, should exhibit similar properties. 

Essentially, an adequate model should capture all information in the data leaving structureless, 

random residuals. If structure remains in the residuals, residual plots often will suggest how to 

modify the model to remove the structure. In this study, plots of residuals versus run sequence 

and a plot of the control mix results revealed a linear trend in the data for each response. 

However, because the run sequence was randomized, this trend had little impact on the fitted 

models. 

1 1 2 3 4 5 19 6 yˆ = − 45 .22 x + 89 .15 x − 3.81x + 1972 x + 38 .36 x + 87 . x

n p

y y

s i i

−

− = ∑ 2 ( ˆ ) (12)

(11)
21

3.5.3 Models for Other Responses 

Using the same procedure described above for 28-day strength, the following models were fit to 

slump (y2), 1-day strength (y3), and 42-day charge passed (y4): 

Linear models were adequate for all responses except 1-day strength, for which the fitted model 

includes 4 quadratic terms that were found to be significant. A natural logarithm transform was 

used to model 42-day charge passed because residual plots showed that the standard deviation of 

the data for charge passed was proportional to the mean. 

3.6 Optimization 

3.6.1 Graphical Interpretation 

Once a valid model is identified, it can be interpreted graphically using response trace plots and 

contour plots. A response trace plot is shown in figure 6. This figure consists of six overlaid 

plots, one for each component. For a given component, the fitted value of the response is plotted 

as the component is varied from its low to high setting in the constrained region, while the other 

components are held in the same relative ratio as a specified reference mixture, here the centroid. 

The plot shows the “effect” of changing each component on 28-day strength. As expected, 

increasing the amount of water decreased strength, while increasing the amount of cement 

increased strength. HRWRA had the largest effect with higher amounts of HRWRA yielding 

higher strength. This may be due to the improved dispersion of the cement and silica fume 

caused by higher amounts of HRWRA. Surprisingly, an increase in silica fume appears to 

reduce strength. However, this apparent reduction may not be significant when compared to the 

underlying experimental error. 

Contour plots are used to identify conditions that give maximum (or minimum) response. 

Because contour plots can only show three components at a time (the others components are set 

at fixed conditions), several must be examined. Figure 7 is a contour plot of 28-day strength for 

water, cement, and HRWRA, with the other components fixed at their centroid values. The plot 

indicates that strength increases rapidly by increasing HRWRA, confirming the result from the 

1 6 2 6 3 4 4 6

3 1 2 3 4 5 6

4214.9 5603.1 43782 44781

ˆ 1209.1 1775.9 74.712 11969 59.589 105.24

x x x x x x x x

y x x x x x x

+ − + +

= − + − − + −

2 1 2 3 4 5 90 6 yˆ = 2166.5x − 2390.5x − 3401.2x + 24268x − 204.83x +169. x

4 1 2 3 4 5 19 6 ln(yˆ ) = 20.82x + 0.629x − 52.33x − 23.40x + 7.24x + 3. x

(13)

(15)

(14)
22

response trace plot. Therefore, in subsequent contour plots, HRWRA will be set at its high 

value. 

Figure 8 shows a contour plot of 28-day strength in water, cement, and silica fume, and figure 9 

shows a contour plot in water, coarse aggregate, and fine aggregate. In each case, HRWRA is 

fixed at its high value, and the other components are fixed at the centroid settings. These plots 

show that strength increases for low water, high cement, low silica fume, low coarse aggregate, 

and high fine aggregate. 

The best overall settings for 28-day strength can be found using the contour plot in figure 10, 

which shows 28-day strength in silica fume, coarse aggregate, and fine aggregate at the best 

settings of water, cement, and HRWRA. The best settings (expressed as volume fractions) are 

water = 0.160, cement = 0.150, silica fume = 0.013, HRWRA = 0.0074, coarse aggregate = 

0.400, and fine aggregate = 0.270, with a predicted strength of 59.53 MPa. 

52

53

54

55

56

57

58

-0.01 0 0.01 0.02 0.03 0.04

Deviation from Ref. Blend (vol. fraction)

28-day strength (MPa)

HRWRA

cement

fine agg

coarse agg

water

silica fume

Figure 6. Response trace plot for 28-day strength (mixture experiment) 
23

Figure 7. Contour plot for 28-day strength in water, cement, and HRWRA 

A: water

0.178

B: cement

0.148

D: HRWRA

0.0225

0.0046

0.160

54.0

56.0

58.0

A: water

B: cement

0.148

D: HRWRA

0.0225

0.130

0.160

52.0

54.0

56.0

58.0

A: water

0.180

B: cement

0.150

C: silica fume

0.033

0.013 0.130

0.160

57.0

57.5

58.0

58.5

A: water

0.180

B: cement

0.150

C: silica fume

0.033

0.130

0.160

56.5

57.0

57.5

58.0

58.5

Figure 8. Contour plot for 28-day strength in water, cement, and silica fume 
24

Figure 10. Contour plot for 28-day strength in silica fume, coarse aggregate, and 

fine aggregate 

Figure 9. Contour plot for 28-day strength in water, coarse aggregate, 

and fine aggregate 

C: silica fume

0.033

0.420

F: Fine agg

0.270

0.250 0.400

0.013

58.25

58.50

58.75

59.00

59.25

2

C: silica fume

0.033

E: Coarse agg

0.420 0.270

0.400

0.013

58.25

58.50

58.75

59.00

59.25

A: water

0.187

E: Coarse agg

0.427

F: Fine agg

0.277

0.250 0.400

0.160

56.0

56.5

57.0

57.5

58.0

58.5

A: water

0.187

0.427 0.277

0.400

0.160

56.0

56.5

57.0

57.5

58.0

58.5
25

3.6.2 Numerical Optimization 

The optimum concrete mix is defined here as that mix which minimizes cost while meeting 

specified performance criteria. Numerical optimization using desirability functions [10] can be 

used to find the optimum mixture proportions in this situation. A desirability function must be 

defined for each response (property). The desirability function takes on values between 0 and 1, 

and may be defined in several ways. Figure 11 shows the desirability functions defined for the 

responses in this experiment. Minimum and maximum specifications are used for strength and 

RCT, resulting in desirability functions with values of 1 above the minimum or below the 

maximum, and 0 otherwise. For example, for 1-day strength the desirability value is 0 below 

22.06 MPa and 1 above 22.06 MPa. At 34.48 MPa, well beyond the maximum value observed in 

the data, the desirability becomes 0. Desirabilities for 28-day strength and 42-day RCT are 

defined similarly. For slump, a range of 50 to 100 mm was specified, but the midpoint of this 

range, 75 mm was selected as the most desirable value (the target value). Therefore, a 

desirability of 1 is given to the target value of 75 mm, with a linear decrease in desirability to a 

value of zero at the lower and upper specification limits (see figure 11). Since cost is to be 

minimized, the desirability function for cost decreases linearly over the range of costs observed 

in the data, as shown in figure 11. 

The individual desirabilities can be expressed mathematically in terms of the predicted response 

values (see equation 9, page 14), which can be calculated for each mixture in the experiment. In 

this study, the overall desirability D was defined as the geometric mean of the individual 

desirability functions di over the feasible region of mixtures [10]: 

The optimal mix is the one with a maximum value of D, which may be determined in several 

ways. The simplest (but crudest) approach is to select the mixture with the largest value of D 

from the design points in the experiment. Another approach is to consider D as a response and 

fit a model to it (as one would fit a model to 28-day strength). A third approach is to use 

Figure 11. Desirability functions for responses in mixture experiment 

D = ( d d d d d )

1/5

1 2 3 4 5 (16)
26

numerical search algorithms to find the maximum value of D [12]. The third approach was used 

here. 

Based on the experimental results, the mix which maximizes D, expressed in volume fractions, is 

water = 0.160, cement = 0.130, silica fume = 0.013, HRWRA = 0.0049, coarse aggregate = 

0.401, and fine aggregate = 0.290, at a cost of $93.01 per m3

. The overall desirability value for 

this mix is 0.964, and the predicted response values are slump = 75 mm, 1-day strength = 

22.07 MPa, 28-day strength = 54.72 MPa, and 42-day RCT value = 631 coulombs. 

3.6.3 Accounting for Uncertainty 

If the fitted functions for each property were known without error, the analysis would be 

complete. However, there is uncertainty in the fitted functions, because they are estimated from 

a sample of data. For example, at the current mix the predicted 1-day target strength is 22.07 

±1.01 MPa. The uncertainty provided is for a 95 percent confidence interval, i.e., we are 95 

percent confident that the interval (21.06, 23.08) contains the true 1-day target strength for this 

mix. If this mix is used, it is quite possible that actual 1-day strength results would fall below the 

minimum acceptable value of 22.06 MPa. Therefore, each specification must be modified to 

account for uncertainties in the fitted functions. The uncertainties in the properties of the current 

best mix can be used to modify the constraints (performance criteria) and identify a revised 

optimal mix based on these new constraints. The revised mix must then be checked to see that 

the original specifications are met. 

The predicted values and 95 percent uncertainties for the remaining responses at the current best 

mix are slump = 75 ±17 mm, 28-day strength = 54.72 ±3.25 MPa, and ln(42-day RCT) = 6.448 

±0.1621

. The modified constraints on the responses which take into account the uncertainties are 

67 mm < slump < 83 mm, 1-day target strength > 23.07 MPa, 28-day target strength > 

54.27 MPa, and ln(42-day RCT) < 6.389 (42-day RCT < 595 coulombs)2

. Repeating the 

numerical optimization, the best mix for this new set of constraints (expressed as volume 

fractions) is water = 0.160, cement = 0.133, silica fume = 0.014, HRWRA = 0.0054, coarse 

aggregate = 0.409, and fine aggregate = 0.2786 at a cost of $96.35. The predicted values and 95 

percent uncertainties for this mix are slump = 76 ±15 mm, 1-day strength = 23.05 ±0.69 MPa, 

28-day strength = 55.20 ±2.31 MPa, and ln(42-day RCT) = 6.404 ±0.114 (corresponding to a 

range of 539 to 678 coulombs). For this mix, the 95 percent confidence intervals for all 

responses meet the original specifications. Accounting for uncertainty increased the cost of the 

optimal mix by about 3 percent ($93.01 to $96.35). The value of optimization can be seen by 

comparing the cost of the optimal mix ($96.35) with the range of costs for all mixtures in the 

experiment ($91.32 to $128.68 per m3

). 

 

1

The predicted value and uncertainty for ln(42-day RCT) are provided because the ln transform was used in the 

model for RCT. The actual 95% confidence interval for 42-day RCT ranges from 537 to 742 coulombs with a 

predicted value of 631 coulombs. 

2

The modified constraints are calculated by adding the uncertainty to the original lower bound constraint, or 

subtracting the uncertainty from the original upper bound. Thus, for slump, the new lower bound is 50 + 17 = 

67 mm, and the new upper bound is 100 – 17 = 83 mm. For 1-day strength, which only has a lower bound, the new 

constraint is 22.06 + 1.01 = 23.07 MPa 
 27

CHAPTER 4 

Laboratory Experiment Using Factorial Approach 

4.1 Introduction 

This chapter describes the application of factorial experiment design to the problem of concrete 

mixture optimization. In a mixture, the total amount (mass or volume) of the product is fixed, 

and the settings of each of the q components are proportions. Because the total amount is 

constrained to sum to one, the component variables are not independent. However, the q

components can be reduced to q-1 mathematically independent variables using the ratio of two 

components as an independent variable [8]. In the case of concrete, w/c is a natural choice for 

this ratio variable. If this strategy is adopted, a factorial approach may be applied. 

4.2 Selection of Materials, Proportions, and Constraints 

The materials used in this experiment were identical to those used in the mixture experiment 

(chapter 3): Type I cement, tap water, #57 crushed limestone, natural sand, silica fume (slurry), 

and naphthalene-sulfonate-based HRWRA. The six components were reduced to five 

independent variables: x1 = w/c (by mass), x2 = fine aggregate volume fraction, x3 = coarse 

aggregate volume fraction, x4 = HRWRA volume fraction, x5 = silica fume volume fraction. 

The volume fraction ranges for coarse aggregate, fine aggregate, HRWRA, and silica fume were 

the same as used in the mixture experiment. The range of w/c (by volume) was calculated from 

the volume fraction limits of the mixture experiment. The lower limit of w/c was 0.16 ÷ 0.15 = 

1.067, and the upper limit was 0.185 ÷ 0.13 = 1.423. These limits were equated to coded limits 

of –1.5 to +1.5 to give the most similar experimental region to that of the mixture experiment. 

Table 9 shows the settings of each variable corresponding to coded values of –1 and +1. 

Table 9. Variable settings corresponding to coded values (factorial experiment) 

For the aggregates, HRWRA, and silica fume, volume fractions were converted to batch masses 

(batch volume for HRWRA) based on total batch size and material properties. The batch masses 

of water and cement were calculated by constraining the sum of volume fractions of all six 

components to sum to one. This gives two equations in two unknowns (the other equation being 

Variable ID Low Setting 

(coded value = -1) 

High Setting 

(coded value = 1) 

w/c ratio x1

1.1263 

(≅ 0.36 by mass) 

1.3637 

(≅ 0.43 by mass) 

Fine aggregate x2 0.2571 0.2853 

Coarse aggregate x3 0.4071 0.4353 

HRWRA x4 0.0051 0.0069 

Silica fume x5 0.0153 0.0247 
 28

the w/c expressed volumetrically) that can be solved for volume fraction of water and volume 

fraction of cement. Entrapped air was ignored in these calculations, although in practice it will 

affect yield calculations and an overall adjustment to the mixture proportions may be necessary 

for proper yield. 

The performance criteria for the mix were as follows: slump between 50 and 100 mm, 1-day 

strength greater than 22 MPa, 28-day strength greater than 51 MPa, and charge passed in the 

RCT less than 700 coulombs. These are the same performance criteria used for the mixture 

experiment, except that the 1-day and 28-day strengths were rounded to 22 and 51 MPa (from 

22.06 and 51.02 MPa used in the mixture experiment)1

. 

4.3 Experiment Design Details 

A central composite design (see chapter 2) was chosen for this experiment. The actual values for 

each variable (expressed in terms of volume fraction) corresponding to the coded levels ±1 are 

shown in table 10. A commercially available software package for experiment design and 

analysis was used to plan the experiment. Thirty-one batches were planned in two nearly 

orthogonal blocks. The first block consisted of a half-fraction of 16 factorial points2

 and 3 center 

points, and the second block consisted of 10 axial points and 2 center points. A total of five 

center points was chosen to allow use of the center point mixes as statistical control mixes to 

assess week-to-week variation, if any, over the five weeks of mixing (in addition to the use of 

center points as replicates to estimate pure error). The run order within each block was 

randomized to reduce the effect of uncontrolled variables. Center point runs were placed first 

and last (based on the total number of runs) with the remainder equally spaced, resulting in three 

center points in the factorial block and two center points in the axial block. The mixture 

proportions for each batch are shown in table 10. 

One batch (run 3) was repeated at the end of the experiment because of suspiciously low strength 

values. A center point batch was included with the repeat to check statistical control. The 

results of the repeated batch were used in subsequent analyses. 

4.4 Specimen Fabrication and Testing 

The materials used in this study included a portland cement conforming to ASTM specification 

C150-94 Type I/II, a #57 crushed limestone coarse aggregate meeting grading requirements of 

AASHTO M43, a natural sand fine aggregate, silica fume (in slurry form), a naphthalenesulfonate based HRWRA (ASTM C494-98 Type F/G), and municipal tap water. Thirty-one 

batches of concrete, each approximately 0.04 m3

 in volume, were prepared over a 6-week period. 

Each batch was prepared in accordance with mixing procedures set forth in ASTM C192-95. 

Table 10. Mixture proportions (per m3) for factorial experiment

 

1

The mixture experiment was originally performed using English units with 1-day strength requirement of 3200 psi 

and 28-day strength requirement of 7400 psi. 
 29

Batch 

(run) 

w/c Fine Agg 

(kg) 

Coarse Agg 

(kg) 

HRWRA 

(L) 

Silica Fume 

(kg) 

Cement 

(kg) 

Water 

(kg) 

 1 0.36 755.7 1098.7 6.90 54.3 408.7 146.1 

 2 0.36 755.7 1174.8 5.10 54.3 369.6 132.2 

 3 0.40 718.4 1136.8 6.00 44.0 394.9 156.1 

 4 0.40 718.4 1136.8 6.00 44.0 394.9 156.1 

 5 0.43 681.0 1098.7 6.90 54.3 405.2 175.4 

 6 0.43 755.7 1098.7 5.10 54.3 370.1 160.2 

 7 0.43 755.7 1098.7 6.90 33.6 380.2 164.6 

 8 0.43 681.0 1098.7 5.10 33.6 420.1 181.9 

 9 0.36 681.0 1098.7 5.10 54.3 453.1 162.0 

 10 0.40 718.4 1136.8 6.00 44.0 394.9 156.1 

 11 0.43 755.7 1174.8 5.10 33.6 345.0 149.4 

 12 0.36 681.0 1098.7 6.90 33.6 464.4 166.0 

 13 0.43 681.0 1174.8 5.10 54.3 370.1 160.2 

 14 0.36 755.7 1098.7 5.10 33.6 425.3 152.1 

 15 0.43 681.0 1174.8 6.90 33.6 380.2 164.6 

 16 0.36 755.7 1174.8 6.90 33.6 380.9 136.2 

 17 0.36 681.0 1174.8 6.90 54.3 408.7 146.1 

 18 0.36 681.0 1174.8 5.10 33.6 425.3 152.1 

 19 0.43 755.7 1174.8 6.90 54.3 330.1 142.9 

 20 0.40 718.4 1212.9 6.00 44.0 355.4 140.5 

 21 0.47 718.4 1136.8 6.00 44.0 357.2 168.1 

 22 0.40 793.1 1136.8 6.00 44.0 355.4 140.5 

 23 0.40 718.4 1136.8 7.80 44.0 392.4 155.1 

 24 0.40 718.4 1136.8 6.00 44.0 394.9 156.1 

 25 0.32 718.4 1136.8 6.00 44.0 441.7 141.3 

 26 0.40 718.4 1136.8 4.20 44.0 397.5 157.1 

 27 0.40 718.4 1136.8 6.00 23.3 408.1 161.3 

 28 0.40 718.4 1060.6 6.00 44.0 434.5 171.7 

 29 0.40 718.4 1136.8 6.00 44.0 394.9 156.1 

 30 0.40 718.4 1136.8 6.00 64.7 381.8 150.9 

 31 0.40 643.7 1136.8 6.00 44.0 434.5 171.7 
 30

Precautions were taken to compensate for mortar retained by the mixer, by “buttering” the mixer 

prior to preparing each batch. The concrete was mixed using a rotating-drum machine mixer with 

a 0.17 m3

 mixing capacity. 

Each batch included sufficient concrete for 2 slump tests, 1 unit weight and air content test 

(ASTM C138 and ASTM C231), and 10 100 mm x 200 mm cylinders. All cylinders were 

fabricated in accordance with ASTM C192, except that external vibration (a vibrating table) was 

utilized for consolidation in lieu of rodding when slump was less than 50 mm. Immediately 

following casting, the cylinders were covered with plastic and left in the molds at room 

temperature for 24 ± 8 hours, after which they were removed from the molds and moist cured at 

23 ± 2 °C until testing. Specimens were tested for compressive strength in accordance with 

ASTM C39 at the ages of 1 and 28 days. In most cases, three cylinders were tested at each age, 

however, where anomalies in either specimen condition or test results were evident, a fourth or 

fifth specimen may have been tested. Prior to compression testing, the ends of each cylinder 

were ground plane and parallel in accordance with ASTM C39 tolerances. Three of the 

remaining cylinders from each batch were used for the RCT testing. Testing was performed 

according to ASTM C1202, except that the 50 mm test specimens were cut from the middle of 

each cylinder instead of from the top. All RCTs were performed at an age of 42 days from 

casting. 

4.5 Results and Analysis 

4.5.1 Responses 

The average values for slump, 1-day strength, 28-day strength, and charge passed (RCT) for each 

batch are shown in table 11, along with the estimated cost (dollars per cubic meter) of the 

concrete mixture. The costs were calculated from the mixture proportions for each batch, based 

on approximate component costs obtained from a local (mid-Atlantic) ready-mix concrete 

producer. Each response was analyzed individually by examining summary plots of the data 

(scatterplots and means plots), fitting a model using ANOVA and least-squares methods, 

validating the model by examining the residuals for trends and outliers, and interpreting the 

model graphically. A detailed discussion of the analysis procedure for 1-day strength is 

presented in the following 2 sections. The analyses for the other responses were performed in a 

similar manner. The models for the other responses are reported in section 4.5.4. 

4.5.2 Exploratory Data Analysis for 1-Day Strength 

One advantage of the factorial approach is the ability to perform an initial assessment of the data 

using graphical techniques such as raw data plots, means plots, scatterplots, and cube plots. 

These techniques are illustrated in figures 12-15 (for complete sets of plots for each response, see 

appendix B). A raw data plot of 1-day strength is shown in figure 12. This plot illustrates the 

variation in the response (1-day strength) over time, for each run. The control batch results, 
 31

Table 11. Test results and costs (factorial experiment) 

Batch 

(run) 

Slump 

(mm) 

1-Day 

(MPa) 

28-Day 

(MPa) 

RCT 

(coul) 

Cost 

($/m3) 

 1 73 16.3 58.5 286 119.77 

 2 44 22.6 59.8 209 113.93 

 3 13 20.8 52.6 160 107.71 

 4 102 16.5 60.4 296 107.71 

 5 57 16.4 55.0 257 118.49 

 6 143 13.6 58.6 541 112.96 

 7 67 12.9 50.4 502 99.21 

 8 13 20.3 52.4 234 97.83 

 9 86 18.4 63.0 305 118.75 

 10 102 15.2 54.8 445 107.71 

 11 140 20.7 62.3 412 93.70 

 12 32 17.1 56.2 252 105.09 

 13 13 24.2 54.2 341 112.98 

 14 76 17.2 50.3 534 99.24 

 15 13 21.3 59.2 278 99.23 

 16 13 20.9 60.5 206 100.27 

 17 57 18.8 56.6 315 119.78 

 18 29 21.9 58.3 355 99.26 

 19 35 16.1 62.9 230 114.36 

 20 38 19.0 58.6 211 105.49 

 21 117 14.5 53.9 458 104.63 

 22 67 17.8 62.4 294 105.48 

 23 64 19.9 67.5 268 111.16 

 24 16 26.4 58.5 189 107.71 

 25 79 19.0 57.0 257 111.52 

 26 64 19.1 50.9 273 104.27 

 27 152 16.8 54.4 705 90.56 

 28 95 20.2 53.3 307 109.93 

 29 35 18.4 55.2 162 107.71 

 30 102 17.4 50.4 332 124.87 

 31 76 18.4 55.2 277 109.95 
 32

shown as hollow squares, give an indication of consistency over time. In this case, the control 

results are all about equal, indicating no time-related effects. Raw data plots are also helpful in 

identifying suspect data values, which may result from (for example) errors in data recording or 

data entry, equipment malfunction, or poor specimen fabrication. 

Figure 13 is a scatterplot showing the effects of varying one factor (w/c) on the 1-day strength. 

In this case, as expected, the 1-day strength decreases with increasing w/c. 

Figure 13. Scatterplot showing effect of w/c on 1-day strength (factorial experiment) 

Raw Data Plot -- 1-day strength

5

10

15

20

25

30

0 5 10 15 20 25 30

Run

1-day strength (MPa)

Figure 12. Raw data plot for 1-day strength (factorial experiment) 

10

15

20

25

30

0.3 0.35 0.4 0.45 0.5

w/c

1-day strength (MPa)
 33

Figure 14 shows a means plot for 1-day strength. The means plot allows comparison of the 

effects of each factor. In this experiment, 1-day strength was clearly influenced by w/c. Other 

factors appear to have had little effect. 

Finally, figure 15 shows a cube plot of 1-day strength with respect to three factors (w/c, fine 

aggregate, and coarse aggregate). A cube plot is a convenient means of assessing quantitative 

effects of three factors on a response. 

Figure 14. Means plot for 1-day strength (factorial experiment) 

Figure 15. Example of cube plot for factorial points 

10

15

20

25

30

Factor

1-day strength (MPa)

w /c SF CA FA HRWRA
 34

4.5.3 Model Fitting and Validation for 1-Day Strength 

After assessing the data graphically, the second step in analysis is to estimate an appropriate 

model for each response. As with the mixture approach, ANOVA and least-squares regression 

techniques are used. The first step is to use ANOVA to determine the appropriate type of model 

(e.g., linear, quadratic). An ANOVA table for sequential model sum of squares, shown in table 

12, suggests that both linear and quadratic terms are significant. A lack-of-fit ANOVA table 

(table 13) suggests that a quadratic model has insignificant lack-of-fit (“Prob > F” = 0.8339). 

Table 12. Sequential model sum of squares for 1-day strength (factorial experiment)

Source Sum of Squares DF Mean Square F Value Prob > F 

 Mean 10780.54 1 10780.54 – – 

 Linear 215.97 5 43.19 22.15 < 0.0001 

 2FI 27.35 10 2.73 1.92 0.1236 

 Quadratic 13.60 5 2.72 3.48 0.0441 

 Cubic (aliased) 1.86 5 0.37 0.31 0.8865 

 Residual 5.95 5 1.19 – – 

 Total 11045.26 31 356.30 – – 

Table 13. Lack-of-fit test for 1-day strength (factorial experiment)

Source Sum of Squares DF Mean Square F Value Prob > F 

 Linear 43.98 21 2.09 1.75 0.3129 

 2FI 16.63 11 1.51 1.27 0.4444 

 Quadratic 3.03 6 0.51 0.42 0.8339 

 Cubic (aliased) 1.18 1 1.18 0.98 0.3772 

 Pure Error 4.78 4 1.19 

When a central composite design is used, the full quadratic model can be estimated, but often 

some of the terms are not significant. The following procedure3

 was used to identify an 

appropriate reduced quadratic model: 

1) Fit the full quadratic model and for each coefficient, calculate the t-statistic for the null 

hypothesis that the coefficient is equal to zero. 

2) Perform the regression again with a partial model containing only those terms that are 

statistically significant (i.e., those terms that had a t-statistic greater than that for the chosen 

 
 35

significance level, in this case 0.05). Calculate the t-statistics again and drop any terms 

which are not significant. 

3) Repeat step 2 until the partial model contains only significant terms. 

4) Add first-order terms required to make the model hierarchical. Hierarchical polynomial 

models make sense under linear transformations such as changing units of temperature from 

Celsius to Fahrenheit [11]. All second-order terms that appear in the model must have 

corresponding first-order terms included in order to make the model hierarchical. 

The ANOVA for the final hierarchical model (with the hierarchical terms shaded) is shown in 

table 14. 

 

Table 14. ANOVA for 1-day strength model (factorial experiment) 

Source Sum of Squares DF Mean Square F Value Prob > F 

Model 240.87 8 30.11 27.76 < 0.0001 

 A 213.26 1 213.26 196.66 < 0.0001 

 B 0.48 1 0.48 0.45 0.5113 

 C 0.04 1 0.04 0.04 0.8433 

 E 2.06 1 2.06 1.90 0.1819 

A2 6.20 1 6.20 5.72 0.0257 

AC 5.15 1 5.15 4.75 0.0404 

AE 7.16 1 7.16 6.60 0.0175 

BC 6.51 1 6.51 6.00 0.0227 

Residual 23.86 22 1.08 – – 

 Lack of fit 19.08 18 1.06 0.89 0.6248 

 Pure error 4.78 4 1.19 – – 

Corr. total 264.72 30 – – – 

For one-day strength, y2, the fitted model was: 

The adequacy of each fitted model was validated quantitatively by calculating statistical 

measures such as residual standard deviation and PRESS, and graphically by examining residual 

plots. The residual standard deviation, s, for this model is 0.99 MPa. A value of s near the 

repeatibility value (replicate standard deviation calculated from center points) is an indication of 

an adequately fitting model. For this experiment, the repeatibility value is 1.04 MPa, which is 

close to s. The PRESS statistic (see page 12) is a measure of how well the model fits each point 

1 5 2 3

1 3

2

2 1 2 3 5 1

3780 3208

ˆ 63.8 860.8 1361.3 450.8 1431.5 323.9 1068

x x x x

y x x x x x x x

+ −

= − − + + − + + (17)
 36

in the design (the smaller the PRESS statistic, the better the fit). PRESS and some other 

quantitative indicators of model adequacy are shown in table 15. 

Table 15. Summary statistics for 1-day strength model (factorial experiment) 

Std. dev. 1.04 

Mean 18.65 

C.V. 5.58 

PRESS 45.16 

R-squared 0.9099

Adjusted R-squared 0.8771

Predicted R-squared 0.8294

The residuals are the deviations of the observed data values from the fitted values, xI , and are 

estimates of the error terms ei in the model. The ei are assumed to be random and normally 

distributed with mean equal to zero and constant standard deviation. A normal probability plot 

of the residuals (shown in figure 16) is used to assess the validity of this assumption. If the error 

terms follow a normal distribution, they will fall on a straight line on the normal probability plot. 

Because they are estimates of the error terms, the residuals should exhibit similar properties. 

If the assumptions are valid, plots of the residuals versus run sequence, predicted values, and 

other independent variables should be random and structureless. If structure remains in the 

residuals, residual plots may suggest modifications to the model that will remove the structure. 

Figure 16. Example of a normal probability plot for model validation 

Residual

Normal % Probability

-2 -1 -0 1 2

1

5

10

20

30

50

70

80

90

95

99

Residual

Normal % Probability

-2 -1 -0 1 2

1

5

10

20

30

50

70

80

90

95

99
 37

Figure 17 shows a plot of residuals versus run sequence for 1-day strength. The plot shows no 

significant structure to the residuals. 

4.5.4 Models for Other Responses 

Using the same procedure described above for 1-day strength, the following models were fit to 

slump (x1), 28-day strength (x3), and 42-day RCT (x4) results: 

4.6 Optimization 

The objective of optimization may be to find the “best settings” (settings which maximize or 

minimize a particular response or responses) or to meet a set of specifications. In either case, 

optimization usually involves considering several responses simultaneously. The same 

(18)

(19)

(20)

3 4

6

1 2

4 5

5

1 1 2 3

20185 1.564 10

ˆ 1365.5 4876.9 8334.7 8256.5 6.698 10 4503.6

x x x x

y x x x x x

+ − ×

= − − − + + × −

3 1 4 5 5 10262 1 5 yˆ = 124.0 − 227.3x + 3390.0x − 3937. x + x x

y x x x x x x x 1 5

2 5

5

6 ˆ 4 = 635.4 + 4445.6 1 −1199.8 2 −1548.5 3 − 31651 5 +1.635×10 −1.448×10

Figure 17. Example of a residual plot (residuals vs. run) for model validation

Run Number

Residuals

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

1 6 11 16 21 26 31

Run Number

Residuals

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

1 6 11 16 21 26 31
 38

optimization strategy that was used in the classical mixture approach can be used for a factorial 

approach (see chapter 3). 

4.6.1 Graphical Optimization 

For three or fewer responses, contour plots can be useful in identifying optimum settings. 

Individual contour plots can be used to identify best settings for each response. Figure 18 shows 

a contour plot of 28-day strength as a function of w/c and silica fume with HRWRA at its middle 

setting. Figure 19 shows the same plot but with HRWRA at the high setting. These plots 

illustrate that the highest strength is reached at the high level of HRWRA coupled with the low 

levels of w/c and silica fume. 

Contour plots can also be overlaid with the constraints for each response defining a subarea of 

settings that meet the response criteria. For example, figure 20 shows all settings meeting the 

following criteria: RCT < 700 and slump equal 50 to 100 mm. The white area in the plot 

indicates the settings meeting the criteria. The gray area on the plot shows the region of settings 

that do not meet the constraints simultaneously. Figure 21 shows the same plot with the added 

constraint that 28-day strength > 51 MPa. As constraints are added, the feasible region usually 

gets smaller. In some situations, no settings will meet all of the criteria. 

Figure 18. 28-day strength in w/c and silica fume (HRWRA at middle setting) 

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

59

58

57

57

58

56

55

54

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

59

58

57

57

58

56

55

54
 39

Figure 19. 28-day strength in w/c and silica fume (HRWRA at high setting)

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

62

61

60

60

59

58

57

56

61

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

62

61

60

60

59

58

57

56

61

Figure 20. Overlay plot for RCT < 700 and slump = 50–100 mm 

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

Slump: 50

Slump: 100

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

Slump: 50

Slump: 100
 40

4.6.2 Numerical Optimization 

For more than three responses considered simultaneously, numerical optimization is often more 

practical than graphical optimization. The numerical optimization technique using desirability 

functions described in chapter 2 and used in the classical mixture experiment (chapter 3) can also 

be applied to the factorial experiment. Desirability functions for this experiment are similar to 

those used in the mixture experiment, except that in some cases (e.g, cost) the endpoint values 

are different. The desirability functions are shown in figure 22. 

Figure 22. Desirability functions for factorial experiment 

Figure 21. Overlay plot for RCT < 700, slump = 50–100 mm, and 28-day strength > 51 MPa

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

Slump: 50

Slump: 100

28-day strength: 51

A: w/c

E: silica fume

0.35 0.37 0.39 0.41 0.43 0.45

0.015

0.017

0.019

0.021

0.023

0.025

Slump: 50

Slump: 100

28-day strength: 51
 41

Numerical optimization gave the following best settings for this concrete mixture: w/c = 0.367 

(by mass), fine aggregate volume fraction = 0.285, coarse aggregate volume fraction = 0.408, 

HRWRA volume fraction = 0.0061, and silica fume volume fraction = 0.0153. The predicted 

response values and associated uncertainties (at a 95 percent confidence level) are slump = 74 

±19 mm, 1-day strength = 22.08 ±1.17 MPa, 28-day strength = 58.65 ±2.32 MPa, and RCT = 

378 ±30 coulombs, at a cost of $100.68 per m3

. 

4.6.3 Accounting for Uncertainty 

As described for the mixture approach, the fitted functions (models) and values predicted from 

them have uncertainty associated with them because they are estimated from data. For example, 

for the optimal mixture proportions shown above the predicted 1-day strength is 22.08 ±1.17 

MPa. If these proportions are used, we can be 95 percent confident that the true 1-day strength 

will lie between 20.91 and 23.25 MPa. But because the specified 1-day strength is 22 MPa, it is 

quite possible that the true 1-day strength will fall below the specified value. Therefore, the 

constraints must be modified to account for the uncertainty in the fitted functions. The 

uncertainties for the optimal mixture can be used to define modified constraints, and a new set of 

optimal mixture proportions can be identified for these new constraints. The predicted responses 

based on the new optimal proportions must be checked to see that the original specifications are 

met. 

The modified constraints are slump between 69 and 81 mm, 1-day strength greater than 23.17 

MPa, 28-day strength greater than 53.32 MPa, and RCT less than 670 coulombs. The best 

mixture for this new set of constraints is w/c = 0.358 (by mass), fine aggregate volume fraction = 

0.282, coarse aggregate volume fraction = 0.4071, HRWRA volume fraction = 0.0062, and silica 

fume volume fraction = 0.0153. The predicted response values and associated uncertainties (at a 

95 percent confidence level) are slump = 74 ±20 mm, 1-day strength = 23.17 ±1.26 MPa, 28-day 

strength = 59.62 ±2.68 MPa, and RCT = 363 ±32 coulombs, at a cost of $101.65 per m3

. All but 

one of the lower or upper bound values for the responses now meet the original specifications. 

The exception is the lower bound for 1-day strength, which is 21.91 MPa (compared with the 

specified value of 22 MPa). In practice, this small difference is probably insignificant; however, 

it may be worthwhile to investigate ways to increase 1-day strength for this mix. A slightly 

lower w/c (0.35, for example) would probably be a sufficient remedy. The predictive models 

estimated from the experiment can be used to predict responses for settings anywhere within the 

experimental design space (i.e., anywhere within the defined variable ranges). However, 

extrapolation beyond the design space is not recommended. 

In the factorial experiment, accounting for uncertainty increased the cost of the optimal mix only 

slightly (from $100.68 to $101.65). As with the mixture experiment, the value of optimization is 

evident when the cost of the optimal mix ($101.65) is compared with the range of mixture costs 

in the experiment ($90.56 to $124.87). 
 
 43

CHAPTER 5 

Development of Interactive Web Site (COST Program) 

5.1 Introduction

The goal of the second phase of this research project was to develop an interactive Web site that 

can be used to optimize concrete mixture proportions using the response surface approach. The 

purpose of this Web site is to introduce this approach to the concrete community and to give 

concrete practitioners an opportunity to apply the approach to their own mixture development. 

Because the response surface approach was likely to be unfamiliar to many practicing engineers 

and producers, the aim was to make it as user-friendly as possible (within budget constraints) and 

to provide as much guidance as possible in interpreting results. 

5.2 Selection of Approach 

A systematic approach is critical when optimizing a HPC mixture subject to several performance 

criteria. The laboratory experiments described in chapters 3 and 4 investigated two such 

approaches: the classical mixture experiment design and the factorial/CCD experiment design. 

Using either of these approaches, a trial batch and testing program can thoroughly examine the 

concrete properties of interest over the selected range of component proportions, and models 

estimated from the data can be used to identify optimal mixes. 

Using a statistical approach to mixture optimization requires a significant investment in trial 

batches and testing. In both the mixture approach and factorial approach, 31 different trial 

batches were required for a 6-component mixture. The large number of runs was required to fit a 

full quadratic model for each response and to provide control runs and replicate runs for 

estimating repeatability. 

If the responses are represented adequately by linear models (as opposed to quadratic), the 

number of trial batches can be reduced by as much as 50 percent. In the mixture experiment 

(chapter 3), linear models were adequate for all but one response (1-day strength). If linear 

model were assumed, the number of experimental runs could have been halved. However, since 

materials and conditions vary by location, the quadratic model is a better initial assumption. 

The factorial approach has an advantage over the mixture approach in that it can be performed 

sequentially (see page 10 of this report). In a sequential approach, the CCD experiment is 

divided into two parts. The adequacy of linear models for the responses can be assessed after 

the initial portion of the experiment (for a 6-component mixture, the first part would consist of 

19 trial batches). If linear response models are sufficient over the range of material proportions 

being considered, the second part of the experiment would not be necessary. If not, the second 

part of the experiment can be run, and quadratic models can be fit to the data. 

In both approaches, the number of runs also can be reduced by holding certain variables 

constant. Reducing the number of components from 6 to 4 would reduce the number of runs in a 

factorial/CCD experiment from 31 to 19. For example, if a user is interested primarily in a 
 44

property that is influenced by cement paste characteristics, he might choose to vary only the 

paste component proportions while holding aggregate constant. 

Based on the experimental results described in chapters 3 and 4, the two approaches, mixture and 

factorial, were evaluated to select the best approach for an interactive Web site. Technical 

suitability and practical considerations (e.g., ease of understanding, ease of use) were considered 

in deciding which method to use for the Web site. While both methods were considered 

technically suitable, the factorial approach was considered to be more practical. The advantage 

of sequential experimentation, which could reduce the number of required trial batches, favored 

the factorial approach. Furthermore, materials engineers are more likely to have encountered 

factorial experiments than mixture experiments, and the factorial approach is more 

straightforward and easier to use, understand, analyze, and interpret. Finally, the statistical 

software to be used for the Web site (DATAPLOT) was better suited for the factorial approach. 

5.3 Considerations in Development 

The following are some of the considerations that shaped the development of the COST software 

and Web site: 

• Introduce the response surface optimization approach in the context of concrete mixture 

proportioning. 

As mentioned in chapter 1, commercially available statistical software packages can provide 

the required experiment design and analysis capabilities needed for this approach. However, 

these packages are not specifically geared toward concrete mixture proportioning. The 

purpose of the COST software is to introduce the RSM approach as a means of optimizing 

concrete mixtures. The COST software is not intended to be a state-of-the-art, all-inclusive, 

“definitive” software product. 

• Minimize the required number of experimental runs needed to produce an optimal mix. 

Concrete producers want to minimize the effort (and cost) needed to identify optimal 

mixtures. Therefore, the maximum number of factors was limited to five, one of which is 

water-cement ratio or water-cementitious materials ratio. The maximum number of 

responses was also limited to five (one of which is cost). 

• Provide flexibility in types of component materials. 

The most common concrete component materials were included, and in each category of 

material (e.g., chemical admixtures, mineral admixtures, aggregates) a “user-defined” 

selection was included to accommodate unusual or new materials. Flexibility was provided 

so a user could, for example, optimize the cementitious matrix alone (i.e., hold aggregates 

constant), or optimize the entire concrete mixture. 
 45

• Recognize the limited statistical background of many potential users. 

For this reason, guidance was included for analysis and interpretation as well as actually 

running the experiment. For example, potential sources of error, the importance of 

randomization, and the importance of accuracy in batching and in following the experimental 

plan are discussed. Because the results and circumstances for any user will vary widely, 

guidance in analysis and interpretation was limited to general aspects, such as indicating 

strong and weak factors. More subtle statistical analysis requires human knowledge and 

judgment. 

• Work within constraints associated with the use of the Web and DATAPLOT software. 

There were several issues to contend with. Speed was one issue—the speed of transfer to 

and from a user’s computer over the Internet to the server housing COST, and the 

computational speed of DATAPLOT. Computations were minimized to reduce waiting 

times. Speed was also an issue in generating graphics (plots). Another limitation associated 

with graphics was the type and quality of graphics that could be generated and presented on 

the Web. DATAPLOT generates postscript plots which were converted to GIF format for 

the Web site. 

Speed issues also prevented the use of a mathematically rigorous numerical optimization 

scheme. Instead, numerical optimization was achieved by calculating a score function at 

each point on a superimposed grid over the range of each factor. For 5 factors and 10 points 

per factor, this requires 105

, or 100,000, calculations. To avoid excessive computation time, 

the grids were limited in size. 

In addition to speed, there were file storage, access, and privacy issues. For example, 

configuration and security constraints require that files be stored on the COST server. They 

cannot be saved on the user’s computer. 

5.4 Description of the Software and Web Site 

5.4.1 Introduction 

COST is an online interactive system developed to assist engineers, concrete producers, and 

researchers in optimizing portland cement concrete mixtures for their particular applications. 

COST applies response surface methodology (RSM), a collection of statistical experiment design 

and analysis methods, to the problem of optimizing concrete mixture proportions. RSM often is 

used in industry for product development, formulation, and improvement, and is applicable to 

problems such as concrete mixture proportioning where several input variables (factors) 

influence a performance measure (response). 

COST is intended to provide an introduction to concrete practitioners who are unfamiliar with 

the concepts and process of applying RSM to concrete mixture proportioning. COST allows 

users to learn how to apply RSM to the problem of optimizing concrete mixtures. 
 46

There are two scenarios for which COST could be applied: 

1. To proportion a concrete mixture to meet a set of specifications at minimum material cost. 

This is probably the most common scenario. 

2. To maximize (or minimize) a particular response or responses, irrespective of cost. 

COST can be used to optimize cement paste, mortar, or concrete mixtures. In all three cases, 

varying the mixture component proportions affects both fresh and hardened properties of the 

paste, mortar, or concrete. The properties (responses) depend on the proportions of the 

components. 

In COST, w/c (or water-cementitious materials ratio, w/cm) is varied along with as many as four 

additional components. These are referred to as variable factors. Other factors may be included 

in the mixture at fixed (constant) levels, and are referred to as fixed factors. The user can 

designate as many as five concrete properties, or responses, (e.g., slump, strength, air content, 

cost, etc.) according to the requirements of the application. 

COST is accessible via the Internet. The program consists of a front-end HTML interface that 

allows the user to enter required information. Underlying code (written in C) processes the 

input, generates the experiment designs and mixture proportions, calls routines for statistical 

analysis, and generates output. The statistical analysis routines are part of an interactive 

statistical software package, DATAPLOT, which was developed at NIST. COST is not intended 

to supplant or compete with commercially available experiment design and analysis software 

packages. Rather, COST’s purpose is to introduce to the concrete practitioner the concepts of 

statistical experiment design and analysis using RSM and to explain how these concepts might 

be applied to concrete mixture proportioning. COST is specifically geared toward applying these 

methods to concrete mixture proportioning. 

This section provides a brief, general overview of COST. The COST User’s Guide, which 

describes the step-by-step approach of the Web site, is included as appendix C of this report. 

5.4.2 Overview of COST Six-Step Process 

The tasks required to optimize a concrete mixture using statistical methods have been assigned to 

the six steps listed below: 

• Specify responses. 

• Specify mixtures. 

• Run trial batches. 

• Enter results. 

• Analyze data. 

• Summarize analysis. 

In most cases, these steps will be performed in the order listed above. Each step is described in 

detail in the COST User’s Guide (see appendix C). 
 47

Before starting the six-step process, the user should define the overall objective of the project. 

Typical objectives include the following: 

• Minimize cost while meeting several performance criteria for responses. 

• Minimize or maximize a single response or several responses. 

Step 1: Specify Responses 

The first step is to specify the responses of interest. Responses are the concrete properties which 

the user is interested in, and are usually dictated by job requirements. The units (e.g., MPa, mm) 

and the allowable range of the response must also be specified. The allowable range defines the 

performance specification for the response. For example, a response like slump may have an 

allowable range between 50 and 100 mm. Another response, like strength, may have a specified 

minimum value greater than 40 MPa. 

Step 2: Specify Mixtures 

Step 2 involves specifying the concrete mixture components and their ranges. Concrete may 

contain a variety of component materials. Allowable material types for this version of COST 

include the following: 

 

• Water.1 

• Cement.1

• Mineral admixtures (up to 4): fly ash, silica fume, slag, other (user specified). 

• Chemical admixtures (up to 3): all user specified. 

• Aggregates (up to 3): coarse, fine, other (user specified). 

Each component, or factor, may be variable or fixed (set at a constant level). For concrete 

mixture proportioning, variable factors would usually be the mixture components expected to 

have the most significant effects on the responses. Fixed factors would be those expected to 

have little or no effect on the responses, allowing them to be held constant in the experiment. 

Any of the components included in COST may be set as variable or fixed; however, COST limits 

the user to a maximum of five variable factors for any one experiment (the greater the number of 

variable factors, the greater the number of trial batches required). Because w/c or w/cm is 

always considered to be one factor, as many as six material components (water, cement, and four 

others) may be varied. 

The user must also provide information about material properties (e.g., for cement, specific 

gravity) and costs for each component to be included. Details on property information required 

can be found in the COST User’s Guide (appendix C). 

 

1

Water and cement are entered in terms of w/c or w/cm. COST always requires that either w/c or w/cm be 

 included as a variable factor. Thus, the two mixture components, water and cement, are accounted for in one 

 factor. 
 48

After the user has decided which factors to include, defined their ranges (for variable factors) or 

constant levels (for fixed factors), and entered required material information into the COST 

program, a trial batch plan is generated. 

Step 3: Running the Experiment 

After generating a trial batch plan, the next step is to perform the experiment. The experiment in 

this case is a set of trial batches from which specimens will be fabricated and tested for the 

responses specified in Step 1. 

Step 4: Enter Results 

After testing is completed, the test results are input into COST for analysis. The data are entered 

into a form, which is set up according to the experimental plan. 

Step 5: Analyze Results 

Analysis of the results consists of 10 tasks, which are performed using one or more statistical 

tools. Table 16 summarizes these analyses. The analysis techniques employed by COST consist 

of both graphical analysis and numerical analysis (modeling), which can be classified in the 

following groups: 

• Quantitative description of data—task 1. 

• Assessing the experiment design—tasks 2, 3, 4. 

• Graphical description of data—tasks 5, 6. 

• Optimization (determining best settings)—tasks 7, 8, 9, 10. 

Examples and details on each analysis task can be found in the COST User’s Guide 

(appendix C). 

Step 6: Summarize Analysis 

The final step summarizes the analysis. The summary includes a list of the component variables, 

the responses, and the optimum settings based on three different perspectives: mean values, 

individual runs, and numerical optimization. A sample of the summary screen is shown in figure 

23. 
 49

Table 16. Summary of analysis tasks and tools in COST 

Task # Task Description Tool(s) 

 1 Characterize response variables Summary statistics 

 2 Assess balance of design Counts plot matrix 

 3 Assess optimality of design points— 

all responses jointly 

Counts in admissible region matrix 

Percentage in admissible region plots 

 4 Assess optimality of design points— 

all four responses jointly Percentage in admissible region plots 

 5 Determine interrelationships between 

responses 

Scatterplots of response variables 

Scatterplots of response variables vs. 

factors 

 6 Assess relationship between 

response variables and factors Means plots of responses vs. factors 

 7 Determine optimal settings for 

each factor 

Best settings based on mean values 

Best settings based on individual runs 

 8 Model fitting and verification Model fitting tool 

 9 Numerical optimization Best settings based on maximum total 

score 

 10 Response prediction Response prediction tool 

Figure 23. Summary screen from COST 
 50

5.5 Future Considerations 

The current version of COST, while functional, is limited in several respects, because it is Webbased software and because of the specific architecture involved. The software runs slowly, the 

graphical capabilities are limited, and data is stored on the host computer instead of the user’s 

computer. A stand-alone, Microsoft® Windows®-based version of COST could be developed in 

the future. However, there are commercially available statistical software packages that could be 

used for this application. Because these packages are general in nature (i.e., not specifically 

geared towards concrete mixture proportioning), some care is needed to assure that they are 

being used correctly. 
51 

ACKNOWLEDGMENTS 

Eric S. Lagergren (Nabisco Research) provided valuable assistance in developing the experiment 

designs for the laboratory experiments and analyzing the experimental results. 

James J. Filliben (NIST) assisted in developing experiment designs for the laboratory work and 

provided experiment design, data analysis and DATAPLOT expertise in the development of the 

COST Web site software. 

Dale Bentz (NIST) programmed the COST Web site software and provided concrete materials 

expertise to the Web site development process. 

Leif Wathne (SaLUT, Inc.) assisted in planning the experiments and supervised the laboratory 

work. 

Joseph Brookman, Brian Goril, and Felix Okoye (formerly of SaLUT, Inc.), performed the 

laboratory work. 

Ken Snyder (NIST) performed rapid chloride testing for Phase I of the laboratory work. 

The “COST User’s Guide” (appendix C) was written by Marcia Simon, Dale Bentz, and Jim 

Filliben.

53 

REFERENCES 

1. Rougeron, P. and P.C. Aitcin, “Optimization of the Composition of a High-Performance 

Concrete,” Cement, Concrete and Aggregates, V. 16, No. 2, December, 1994, pp. 115-124. 

2. ACI Committee 211, “Standard Practice for Selecting Proportions for Normal, 

Heavyweight, and Mass Concrete.” In ACI Manual of Concrete Practice, Volume 1. 

Detroit: American Concrete Institute, 1995. 

3. Myers, R.H. and D.C. Montgomery. Response Surface Methodology: Process and Product 

Optimization Using Designed Experiments, New York: Wiley, 1995. 

4. Luciano, J.J, Nmai, C.K., and J.R. DelGado, “A Novel Approach to Developing HighStrength Concrete,” Concrete International, May, 1991, pp. 25-29. 

5. Bajorski, P., Streeter, D.A., and R.J. Perry, “Applying Statistical Methods for Further 

Improvement of High-Performance Concrete for New York State Bridge Decks,” 

Transportation Research Record No. 1574, Transportation Research Board, Washington, 

DC, 1997, pp. 71-79. 

6. DeLarrard, F., and A. Belloc, “The Influence of Aggregate on the Compressive Strength of 

Normal and High-Strength Concrete.” ACI Materials Journal, V. 94, No. 5, 1997, 

pp. 417-426. 

7. Cornell, How to Run Mixture Experiments for Product Quality, American Society for 

Quality Control, 1990. 

8. Piepel and Cornell, “Mixture Experiment Approaches: Examples, Discussion, and 

Recommendations.” Journal of Quality Technology, Vol. 26, No. 3, July, 1994, 

pp. 177-196. 

9. Box, G.E.P, Hunter, W.G., and J.S. Hunter, Statistics for Experimenters, New York: Wiley, 

1978. 

10. Derringer, G. and R. Suich, “Simultaneous Optimization of Several Response Variables,” 

Journal of Quality Technology, Vol. 12, No. 4, October, 1980, pp. 214-219.

11. Peixoto, J. L., “Hierarchical Variable Selection in Polynomial Regression Models,” The 

American Statistician, Vol. 41, No. 4, November, 1987. 

12. Design Expert 6 User’s Guide, Stat-Ease, Inc., Minneapolis, MN, 2000. 

 A-1

APPENDIX A. Experiment Design and Data Analysis for Mixture Experiment 

A.1 Experiment Design and Response Data 

Table A-1. Mixture experiment design in terms of volume fractions of components 

Standard 

Order 

Design 

ID 

Run 

Order 

A 

Water 

B 

Cement

C 

Silica Fume

D 

HRWRA

E 

Coarse Agg 

F 

Fine Agg 

 35 127 1 0.1720 0.1395 0.0130 0.0060 0.4098 0.2598

 7 15 2 0.1656 0.1500 0.0270 0.0074 0.4000 0.2500

 14 38 3 0.1767 0.1417 0.0270 0.0046 0.4000 0.2500

 11 22 4 0.1850 0.1474 0.0130 0.0046 0.4000 0.2500

 22 71 5 0.1600 0.1500 0.0130 0.0074 0.4098 0.2598

 4 11 6 0.1850 0.1300 0.0130 0.0046 0.4174 0.2500

 2 5 7 0.1600 0.1300 0.0270 0.0046 0.4284 0.2500

 8 16 8 0.1600 0.1300 0.0130 0.0046 0.4424 0.2500

 27 91 9 0.1850 0.1300 0.0130 0.0074 0.4073 0.2573

 29 101 10 0.1712 0.1500 0.0130 0.0046 0.4112 0.2500

 24 78 11 0.1850 0.1300 0.0270 0.0074 0.4003 0.2503

 37 127 12 0.1720 0.1395 0.0130 0.0060 0.4098 0.2598

 9 20 13 0.1850 0.1300 0.0130 0.0046 0.4000 0.2674

 30 103 14 0.1600 0.1500 0.0130 0.0060 0.4210 0.2500

 5 13 15 0.1600 0.1300 0.0130 0.0046 0.4000 0.2924

 12 28 16 0.1600 0.1300 0.0270 0.0046 0.4000 0.2784

 28 98 17 0.1600 0.1400 0.0130 0.0046 0.4324 0.2500

 36 127 18 0.1720 0.1395 0.0130 0.0060 0.4098 0.2598

 26 89 19 0.1600 0.1400 0.0130 0.0046 0.4000 0.2824

 39 163 20 0.1656 0.1343 0.0165 0.0067 0.4234 0.2536

 31 110 21 0.1712 0.1500 0.0130 0.0046 0.4000 0.2612

 1 5 22 0.1600 0.1300 0.0270 0.0046 0.4284 0.2500

 3 11 23 0.1850 0.1300 0.0130 0.0046 0.4174 0.2500

 25 87 24 0.1600 0.1300 0.0130 0.0046 0.4212 0.2712

 20 66 25 0.1600 0.1300 0.0270 0.0074 0.4128 0.2628

 16 38 26 0.1767 0.1417 0.0270 0.0046 0.4000 0.2500

 18 63 27 0.1725 0.1300 0.0270 0.0074 0.4131 0.2500

 17 48 28 0.1725 0.1300 0.0130 0.0074 0.4000 0.2771

 21 70 29 0.1600 0.1500 0.0130 0.0060 0.4000 0.2710

 13 37 30 0.1600 0.1400 0.0270 0.0074 0.4000 0.2656

 19 65 31 0.1600 0.1400 0.0270 0.0074 0.4156 0.2500

 38 127 32 0.1720 0.1395 0.0130 0.0060 0.4098 0.2598

 32 116 33 0.1725 0.1300 0.0270 0.0074 0.4000 0.2631

 10 20 34 0.1850 0.1300 0.0130 0.0046 0.4000 0.2674

 23 71 35 0.1600 0.1500 0.0130 0.0074 0.4098 0.2598

 33 123 36 0.1600 0.1400 0.0200 0.0060 0.4120 0.2620

 34 127 37 0.1720 0.1395 0.0130 0.0060 0.4098 0.2598

 6 15 38 0.1656 0.1500 0.0270 0.0074 0.4000 0.2500

 15 38 39 0.1767 0.1417 0.0270 0.0046 0.4000 0.2500
 A-2

Table A-2. Mixture experiment: slump and 1-day strength data 

 

Obs Design ID Run 

Order 

Slump 

(mm) 

1-Day Strength 

(MPa) 

 32 127 1 50.8 54.0 24.3 24.4 24.0 –

 6 15 2 25.4 31.8 24.6 22.9 25.9 –

 13 38 3 19.1 19.1 22.7 22.6 22.1 –

 10 22 4 69.9 63.5 21.1 21.4 21.8 – 

 20 71 5 63.5 50.8 26.9 27.2 26.5 27.5 

 3 11 6 108.0 95.3 16.7 16.6 17.0 – 

 1 5 7 12.7 12.7 21.7 23.3 22.2 – 

 7 16 8 38.1 31.8 21.8 21.6 21.4 – 

 25 91 9 203.2 196.9 17.1 16.6 16.7 – 

 27 101 10 25.4 19.1 26.4 26.5 26.8 – 

 22 78 11 127.0 127.0 19.2 19.3 19.0 – 

 33 127 12 101.6 95.3 20.8 21.9 21.9 – 

 8 20 13 114.3 120.7 18.3 18.0 18.3 – 

 28 103 14 63.5 63.5 28.0 26.8 27.3 – 

 5 13 15 63.5 50.8 22.7 21.7 21.0 – 

 11 28 16 31.8 25.4 21.1 22.9 22.5 – 

 26 98 17 31.8 31.8 25.2 26.4 24.2 – 

 34 127 18 95.3 88.9 21.5 21.6 23.8 – 

 24 89 19 38.1 38.1 18.1 23.1 24.0 – 

 36 163 20 95.3 95.3 22.1 21.6 22.4 – 

 29 110 21 50.8 50.8 24.5 25.1 24.4 – 

 2 5 22 25.4 25.4 22.1 24.0 24.2 – 

 4 11 23 114.3 114.3 17.1 16.5 15.9 – 

 23 87 24 63.5 69.9 24.9 20.6 23.2 – 

 18 66 25 76.2 76.2 25.0 24.1 25.1 – 

 14 38 26 31.8 25.4 22.9 23.6 22.5 – 

 16 63 27 101.6 95.3 21.6 22.0 21.4 – 

 15 48 28 177.8 165.1 22.7 22.7 23.7 – 

 19 70 29 50.8 50.8 27.5 27.3 27.6 – 

 12 37 30 38.1 31.8 26.8 27.8 27.4 – 

 17 65 31 31.8 31.8 28.6 25.5 27.6 – 

 35 127 32 120.7 120.7 22.9 22.2 22.1 – 

 30 116 33 114.3 114.3 23.9 23.7 24.1 – 

 9 20 34 127.0 127.0 18.8 18.6 18.5 – 

 21 71 35 114.3 101.6 29.5 28.0 28.9 – 

 31 123 36 76.2 69.9 26.2 26.4 27.3 – 

 32 127 37 101.6 101.6 23.9 24.3 24.2 – 

 6 15 38 50.8 50.8 29.2 27.6 29.5 – 

 13 38 39 25.4 25.4 23.3 25.4 22.2 – 
 A-3

Table A-3. Mixture experiment: 28-day strength and RCT charge passed data 

Obs Design ID Run 

 Order 

28-Day Strength 

(psi) 

RCT Charge Passed 

(coulombs) 

 32 127 1 51.0 52.5 54.9 – – – – 

 6 15 2 59.4 58.7 59.5 – – – – 

 13 38 3 50.8 50.1 52.7 – – – – 

 10 22 4 47.9 48.0 48.8 – 1203 1310 1321 

 20 71 5 51.9 56.7 57.0 – 901 790 894 

 3 11 6 47.3 51.7 46.6 – 1141 1308 1038 

 1 5 7 46.1 49.4 49.9 – 422 352 – 

 7 16 8 54.6 53.2 51.5 – 708 – 843 

 25 91 9 58.8 60.4 62.1 – 1092 1113 877 

 27 101 10 57.0 52.3 51.6 – 730 736 767 

 22 78 11 51.0 55.4 48.6 – 474 454 549 

 33 127 12 51.1 52.3 47.4 – 853 885 789 

 8 20 13 51.9 50.2 50.7 – 995 922 793 

 28 103 14 60.4 56.6 46.2 55.1 607 576 565 

 5 13 15 51.0 55.6 52.8 – 575 719 758 

 11 28 16 56.5 50.8 53.4 – 327 268 282 

 26 98 17 55.5 50.5 49.8 – 580 579 653 

 34 127 18 53.7 54.2 54.3 – 841 852 848 

 24 89 19 52.6 58.3 50.2 56.3 677 656 826 

 36 163 20 61.1 61.0 60.2 – 544 552 566 

 29 110 21 52.3 52.9 54.4 – 716 804 857 

 2 5 22 56.3 52.0 53.9 – 308 441 296 

 4 11 23 49.2 47.8 46.9 – 894 1054 956 

 23 87 24 53.0 50.4 49.4 – 751 618 732 

 18 66 25 57.1 62.6 59.7 – 326 319 303 

 14 38 26 57.4 51.8 50.3 – 450 346 375 

 16 63 27 56.6 54.5 54.5 – 324 324 257 

 15 48 28 58.9 58.0 57.3 – 622 702 723 

 19 70 29 54.7 52.5 56.4 – 496 494 524 

 12 37 30 49.2 57.8 59.7 57.4 247 254 234 

 17 65 31 53.5 49.1 50.8 – 285 350 296 

 35 127 32 57.8 58.6 55.0 – 661 567 680 

 30 116 33 57.8 57.3 53.5 – 367 358 343 

 9 20 34 51.1 51.9 51.9 – 804 901 754 

 21 71 35 64.9 65.6 65.2 – 550 566 543 

 31 123 36 61.5 59.8 61.8 – 318 312 390 

 32 127 37 53.8 55.4 54.6 – 640 614 665 

 6 15 38 61.6 57.5 55.2 – 235 232 250 

 13 38 39 54.5 53.8 55.4 – 323 379 293 
 A-4

A.2 Data Analysis and Model Fitting 

A.2.1 Slump 

Table A-4. Mixture experiment: sequential model sum of squares for slump 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 212697.8 1 212697.8 – – 

Linear 62543.72 5 12508.74 45.64 < 0.0001 

Quadratic 4107.07 15 273.80 1.00 0.5016 

Special cubic (aliased) 2164.86 7 309.27 1.27 0.3703 

Cubic (aliased) 0.00 0 – – 

Residual 1950.91 8 243.86 – – 

Total 283464.3 36 7874.01 – – 

Table A-5. Mixture experiment: lack-of-fit test for slump 

Source Sum of Squares DF Mean Square F Value Prob > F

Linear 6271.93 22 285.09 1.17 0.4335 

Quadratic 2164.86 7 309.27 1.27 0.3703 

Special cubic (aliased) 0.00 0 – – – 

Cubic (aliased) 0.00 0 – – – 

Pure error 1950.91 8 243.86 – – 

Table A-6. Mixture experiment: model summary statistics for slump 

Source Std. Dev. r2 Adj. r2

 Pred. r2 PRESS 

Linear 16.56 0.8838 0.8644 0.8407 11272.43 

Quadratic 16.56 0.9418 0.8643 0.6049 27956.70 

Special cubic (aliased) 15.62 0.9724 0.8794 – undefined 

Cubic (aliased) – – – – undefined 
 A-5

Table A-7. Mixture experiment: ANOVA for slump mixture model 

Source Sum of Squares DF Mean Square F Value Prob > F

Model 62543.72 5 12508.74 45.64 < 0.0001

Linear mixture 62543.72 5 12508.74 45.64 < 0.0001

Residual 8222.84 30 274.09 – – 

Lack of fit 6271.93 22 285.09 1.17 0.4335

Pure error 1950.91 8 243.86 – – 

Corrected total 70766.56 35 – – – 

Table A-8. Mixture experiment: coefficient estimates for slump mixture model 

Component Coeff. 

Estimate DF Std. Error 95% CI Low 

95% CI 

High 

Water 155.68 1 10.12 135.03 176.34 

Cement –37.53 1 13.34 –64.77 –10.30 

Silica fume –80.39 1 17.95 –117.04 –43.74 

HRWRA 1092.78 1 100.02 888.51 1297.04 

Coarse aggregate 55.14 1 8.85 37.07 73.22 

Fine aggregate 71.03 1 9.29 52.06 90.00 

Table A-9. Mixture experiment: adjusted effects for slump mixture model 

Component Adjusted 

Effect DF Std. Error Approx. t for HoEffect = 0 Prob > t 

Water –138.04 1 12.96 –2.94 0.0063 

Cement –139.80 1 12.43 –11.25 < 0.0001 

Silica fume –114.84 1 10.42 –11.02 < 0.0001 

HRWRA 70.00 1 6.83 10.25 < 0.0001 

Coarse aggregate –185.17 1 20.68 –8.95 < 0.0001 

Fine aggregate –166.11 1 22.02 –7.54 < 0.0001 

Model equation in terms of pseudocomponents:

 Slump = 155.68*A – 37.53*B – 80.39*C + 1092.78*D + 55.14*E + 71.03*F 

 Model equation in terms of real components:

 Slump = 2166.5*water – 2390.5*cement – 3401.2*silica fume + 24267.7*HRWRA 

 – 204.8*Coarse agg +169.9*Fine agg 
 A-6

Figure A-1. Mixture experiment: normal probability plot for slump 

Figure A-2. Mixture experiment: residuals vs. run for slump 

Studentized Residuals

Normal % Probability

-1.82 -0.61 0.61 1.82 3.04

1

5

10

20

30

50

70

80

90

95

99

Run Number

Studentized Residuals

-3.00

-1.49

0.02

1.53

3.04

4 9 14 19 24 29 34 39
 A-7

Figure A-3. Mixture experiment: Cook's distance for slump 

Figure A-4. Mixture experiment: trace plot for slump 

Run Number

Cook's Distance

0.00

0.03

0.06

0.09

0.12

0.15

4 9 14 19 24 29 34 39

Deviation from Reference Blend

Slump

-0.238 -0.053 0.132 0.317 0.502

13

60

106

153

200

water

silica fume

HRWRA

coarse agg

fine agg

cement
 A-8

Figure A-5. Mixture experiment: contour plot for slump in water, cement, and silica fume

Figure A-6. Mixture experiment: contour plot of slump in water, cement, and 

HRWRA 

A: water

0.182

B: cement

0.152

C: silica fume

0.035

0.013 0.130

0.160

40

60

80

100

120

A: water

0.178

B: cement

0.148

D: HRWRA

0.0225

0.0046 0.130

0.160

40

60

80

100

120

140
 A-9

A.2.2 1-Day Strength 

Table A-10. Mixture experiment: sequential model sum of squares for 1-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 19278.40 1 19278.40 – – 

Linear 351.33 5 70.27 57.34 < 0.0001 

Quadratic 27.14 15 1.81 2.82 0.0266 

Special cubic (aliased) 3.43 7 0.49 0.63 0.7199 

Cubic (aliased) 0.00 0 – – – 

Residual 6.19 8 0.77 – – 

Total 19666.48 36 546.29 – – 

Table A-11. Mixture experiment: lack-of-fit test for 1-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F

Linear 30.57 22 1.39 1.79 0.1992 

Quadratic 3.43 7 0.49 0.63 0.7199 

Special cubic (aliased) 0.000 0 – – – 

Cubic (aliased) 0.000 0 – – – 

Pure error 6.19 8 0.77 – – 

Table A-12. Mixture experiment: summary statistics for 1-day strength 

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS 

Linear 1.11 0.9053 0.8895 0.8611 53.89 

Quadratic 0.80 0.9752 0.9421 0.8566 55.65 

Special cubic (aliased) 0.88 0.9840 0.9302 – undefined 

Cubic (aliased) – – – – undefined 
 A-10

Table A-13. Mixture experiment: ANOVA for 1-day strength mixture model

Source Sum of 

Squares DF Mean Square F Value Prob > F 

Model 373.09 9 41.45 71.84 < 0.0001 

Linear mixture 351.77 5 70.35 121.93 < 0.0001 

AF 4.14 1 4.14 7.18 0.0126 

BF 5.38 1 5.38 9.32 0.0052 

CD 3.96 1 3.96 6.86 0.0145 

DF 7.83 1 7.83 13.58 0.0011 

Residual 15.00 26 0.58 – – 

Lack of fit 8.81 18 0.49 0.63 0.8010 

Pure error 6.19 8 0.77 – – 

Corrected total 388.09 35 – – – 

Table A-14. Mixture experiment: coefficient estimates for 1-day strength mixture model 

Component Coeff. 

Estimate DF Std. Error 95% CI Low 

95% CI 

High 

water 12.93 1 0.62 11.64 14.21 

Cement 35.42 1 0.83 33.72 37.12 

Silica fume 24.89 1 1.15 22.52 27.25 

HRWRA 10.83 1 8.88 -7.42 29.07 

Coarse aggregate 22.04 1 0.43 21.16 22.93 

Fine aggregate 21.50 1 0.57 20.32 22.67 

Water—fine aggregate 7.58 1 2.83 1.77 13.39 

Cement—fine aggregate -10.07 1 3.30 -16.86 -3.29 

Silica fume—HRWRA 78.71 1 30.04 16.95 140.47 

HRWRA—fine aggregate 80.50 1 21.85 35.60 125.41 

 Model equation in terms of pseudocomponents: 

Slump = 12.93*A + 35.42*B + 24.89*C + 10.83*D + 22.04*E + 21.50*F + 7.58* A*F 

 – 10.07*B*F + 78.71*C*D + 80.50*D*F 

Model equation in terms of real components:

Slump = –1209.1*water + 1775.9*cement – 74.71*SF – 11969*HRWRA 

 + 59.59*Coarse agg – 105.24*Fine agg + 4214.9*water*Fine agg 

 – 5603.1*cement*Fine agg + 43782 *SF*HRWRA + 44781*HRWRA*Fine agg 
 A-11

Figure A-7. Mixture experiment: normal probability plot for 1-day strength

Figure A-8. Mixture experiment: residuals vs. run for 1-day strength 

Studentized Residuals

Normal % Probability

-2.01 -0.97 0.08 1.12 2.17

1

5

10

20

30

50

70

80

90

95

99

Run Number

Studentized Residuals

-3.00

-1.50

0.00

1.50

3.00

4 9 14 19 24 29 34 39
 A-12

Figure A-9. Mixture experiment: trace plot for 1-day strength 

Figure A-10. Mixture experiment: contour plot of 1-day strength in 

water, cement, and silica fume

A: water

0.182

B: cement

0.152

C: silica fume

0.035

0.013 0.130

0.160

20.0

21.0

22.0

23.0

24.0

25.0

26.0

27.0

Deviation from Reference Blend

1 day strength

-0.238 -0.053 0.132 0.317 0.502

16.5

19.6

22.7

25.7

28.8

water

cement

silica fume HRWRA

coarse agg

fine agg
 A-13

Figure A-11. Mixture experiment: contour plot of 1-day strength in water, 

cement, and HRWRA 

Figure A-12. Mixture experiment: contour plot of 1-day strength in water, 

cement, and fine aggregate 

A: water

0.183

B: cement

0.153

D: HRWRA

0.0274

0.0046 0.130

0.160

20

21

22

23

24

25

26

27

A: water

0.185

B: cement

0.155

F: Fine agg

0.275

0.250 0.130

0.160

19

20

21

22

23

24

25 26 27

28
 A-14

Figure A-13. Mixture experiment: contour plot of 1-day strength in silica fume, 

HRWRA, and fine aggregate 

Figure A-14. Mixture experiment: contour plot of 1-day strength in silica 

fume, coarse aggregate, and fine aggregate 

C: silica fume

0.029

D: HRWRA

0.0210

F: Fine agg

0.266

0.250 0.0046

0.013

27.0

27.5

28.0

28.5

29.0

29.5

30.0

C: silica fume

0.033

E: Coarse agg

0.420

F: Fine agg

0.270

0.250 0.400

0.013

28.0

28.5

29.0

29.5

30.0
 A-15

A.2.3 28-Day Strength 

 

Table A-15. Mixture experiment: sequential model sum of squares for 28-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 106213.0 1 106213.0 – – 

Linear 257.52 5 51.50 5.46 0.0011 

Quadratic 135.19 15 9.01 0.92 0.5665 

Special cubic (aliased) 55.45 7 7.92 0.69 0.6826 

Cubic (aliased) 0.00 0 – – – 

Residual 92.17 8 11.52 – – 

Total 106753.3 36 2965.37 – – 

Table A-16. Mixture experiment: lack-of-fit test for 28-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F

Linear 190.64 22 8.67 0.75 0.7193 

Quadratic 55.45 7 7.92 0.69 0.6826 

Special cubic (aliased) 0.00 0 – – – 

Cubic (aliased) 0.00 0 – – – 

Pure error 92.17 8 11.52 – – 

Table A-17. Mixture experiment: model summary statistics for 28-day strength 

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS 

Linear 3.07 0.4755 0.3894 0.2678 395.63 

Quadratic 3.14 0.7268 0.3625 –1.5516 1378.72 

Special cubic (aliased) 3.39 0.8294 0.2537 – undefined 

Cubic (aliased) – – – – undefined 
 A-16

Table A-18. Mixture experiment: ANOVA for 28-day strength mixture model 

Source Sum of Squares DF Mean Square F Value Prob > F 

Model 257.52 5 51.50 5.46 0.0011 

Linear mixture 257.52 5 51.50 5.46 0.0011 

Residual 282.81 30 9.43 – – 

Lack of fit 190.64 22 8.67 0.75 0.7193 

Pure error 92.17 8 11.52 – – 

Corrected total 540.33 35 – – – 

Table A-19. Mixture experiment: estimated coefficients for 28-day strength mixture model 

Component Coeff. 

Estimate DF Std. Error 95% CI Low 

95% CI 

High 

Water 48.60 1 1.88 44.77 52.43 

Cement 54.30 1 2.47 49.25 59.35 

Silica fume 50.36 1 3.33 43.56 57.15 

HRWRA 134.13 1 18.55 96.24 172.01 

Coarse aggregate 52.14 1 1.64 48.79 55.50 

Fine aggregate 54.21 1 1.72 50.70 57.73 

Table A-20. Mixture experiment: adjusted effects for 28-day strength mixture model 

Component Adjusted 

Effect DF Std. Error Approx. t for Ho Effect = 0 Prob > t 

Water -12.04 1 2.40 -5.01 < 0.0001 

Cement -6.41 1 2.31 -2.78 0.0093 

Silica fume -6.05 1 1.93 -3.13 0.0039 

HRWRA 5.43 1 1.27 4.28 0.0002 

Coarse aggregate -16.17 1 3.84 -4.22 0.0002 

Fine aggregate -13.69 1 4.08 -3.35 0.0022 

Model equation in terms of pseudocomponents:

 28-Day Strength = 48.60*A + 54.30*B + 50.36*C + 134.13*D + 52.14*E + 54.21*F 

Model equation in terms of real components:

28-Day Strength = –45.22*water + 89.15*cement – 3.809*silica fume + 1972*HRWRA 

 + 38.36*Coarse agg + 87.19*Fine agg 
 A-17

Figure A-15. Mixture experiment: normal probability plot for 28-day strength

Figure A-16. Mixture experiment: residuals vs. run for 28-day strength 

Studentized Residuals

Normal % Probability

-2.23 -1.12 -0.01 1.11 2.22

1

5

10

20

30

50

70

80

90

95

99

Run Number

Studentized Residuals

-3.00

-1.50

0.00

1.50

3.00

4 9 14 19 24 29 34 39
 A-18

Figure A-17. Mixture experiment: Cook’s distance vs. run for 28-day strength 

Figure A-18. Mixture experiment: trace plot for 28-day strength 

Run Number

Cook's Distance

0.00

0.05

0.10

0.15

0.20

4 9 14 19 24 29 34 39

Deviation from Reference Blend

28 day strength

-0.238 -0.053 0.132 0.317 0.502

52.0

53.5

55.0

56.5

58.0

water

cement

silica fume

HRWRA

coarse agg

fine agg
 A-19

Figure A-19. Mixture experiment: contour plot of 28-day strength in water, silica 

fume, and HRWRA 

Figure A-20. Mixture experiment: contour plot of 28-day strength in water, silica 

fume, and coarse aggregate 

A: water

0.175

C: silica fume

0.028

D: HRWRA

0.0198

0.0046 0.013

0.160

52

53

54

55

56

57

A: water

0.183

C: silica fume

0.036

E: Coarse agg

0.423

0.400 0.013

0.160

56.75

57.00

57.25

57.50

57.75

58.00

58.25
 A-20

Figure A-21. Mixture experiment: contour plot of 28-day strength in cement, coarse 

aggregate, and fine aggregate 

B: cement

0.170

E: Coarse agg

0.440

F: Fine agg

0.290

0.250 0.400

0.130

57.75

58.00

58.25

58.50

58.75 59.00

59.25
 A-21

A.2.4 RCT Charge Passed 

Table A-21. Mixture experiment: sequential model sum of squares for RCT charge passed 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 1441.26 1 1441.26 – – 

Linear 6.75 5 1.35 57.33 < 0.0001 

Quadratic 0.38 15 0.03 1.17 0.3846 

Special cubic (aliased) 0.11 7 0.02 0.57 0.7607 

Cubic (aliased) 0.00 0 – – – 

Residual 0.22 8 0.03 – – 

Total 1448.72 36 40.24 – – 

Table A-22. Mixture experiment: lack-of-fit test for RCT charge passed 

Source Sum of Squares DF Mean Square F Value Prob > F

Linear 0.490 22 0.022 0.82 0.6666 

Quadratic 0.109 7 0.016 0.57 0.7607 

Special cubic (aliased) 0.000 0 – – – 

Cubic (aliased) 0.000 0 – – – 

Pure error 0.217 8 0.027 – – 

Table A-23. Mixture experiment: model summary statistics for RCT charge passed 

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS 

Linear 0.15 0.9053 0.8895 0.8647 1.01 

Quadratic 0.15 0.9563 0.8980 0.8044 1.46 

Special cubic (aliased) 0.16 0.9709 0.8726 – undefined 

Cubic (aliased) – – – – undefined 

Table A-24. Mixture experiment: ANOVA for RCT charge passed mixture model 

Source Sum of Squares DF Mean Square F Value Prob > F

Model 6.75 5 1.35 57.33 < 0.0001 

Linear mixture 6.75 5 1.35 57.33 < 0.0001 

Residual 0.71 30 0.024 – – 

Lack of fit 0.49 22 0.022 0.82 0.6666 

Pure error 0.22 8 0.027 – – 

Corrected total 7.46 35 – – – 
 A-22

Table A-25. Mixture experiment: estimated coefficients for RCT charge passed mixture 

model 

Component Coeff. 

Estimate DF Std. Error 95% CI Low 

95% CI 

High 

 Water 7.2 1 0.094 7.01 7.39 

 Cement 6.34 1 0.12 6.09 6.60 

 Silica fume 4.10 1 0.17 3.76 4.44 

 HRWRA 5.32 1 0.93 3.43 7.22 

 Coarse aggregate 6.62 1 0.082 6.46 6.79 

 Fine aggregate 6.45 1 0.086 6.28 6.63 

Table A-26. Mixture experiment: adjusted effects for RCT charge passed mixture model 

Component Adjusted 

Effect DF Std. Error Approx. t for HoEffect = 0 Prob > t 

 Water 0.84 1 0.12 7.03 < 0.0001 

 Cement 0.19 1 0.12 1.65 0.1087 

 Silica fume -0.76 1 0.097 -7.83 < 0.0001 

 HRWRA -0.054 1 0.063 -0.85 0.3999 

 Coarse aggregate 0.74 1 0.19 3.86 0.0006 

 Fine aggregate 0.53 1 0.20 2.62 0.0138 

Model equation in terms of pseudocomponents:

ln(RCT charge passed) = 7.20*A + 6.34*B + 4.10*C + 5.32*D + 6.62*E + 6.45*F 

Model equation in terms of real components:

ln(RCT charge passed) = 20.82*water + 0.629*cement - 52.33*silica fume – 23.41*HRWRA 

 + 7.235*Coarse agg + 3.190*Fine agg 
 A-23

Figure A-22. Mixture experiment: normal probability plot for RCT charge passed 

(no transform) 

Figure A-23. Mixture experiment: normal probability plot for RCT charge 

passed (natural log transform) 

Studentized Residuals

Normal % Probability

-1.37 -0.24 0.89 2.02 3.15

1

5

10

20

30

50

70

80

90

95

99

Studentized Residuals

Normal % Probability

-1.73 -0.89 -0.06 0.78 1.61

1

5

10

20

30

50

70

80

90

95

99
 A-24

Figure A-24. Mixture experiment: residuals vs. predicted for RCT charge 

passed (no transform) 

Figure A-25. Mixture experiment: residuals vs. run for RCT charge 

passed (no transform) 

Predicted

Studentized Residuals

-3.00

-1.50

0.00

1.50

3.00

100 300 500 700 900

Run Number

Studentized Residuals

-3.00

-1.46

0.08

1.61

3.15

4 9 14 19 24 29 34 39
 A-25

Figure A-26. Mixture experiment: residuals vs. predicted for RCT charge 

passed (natural log transform) 

Figure A-27. Mixture experiment: residuals vs. run for RCT charge 

passed (natural log transform) 

Predicted

Studentized Residuals

-3.00

-1.50

0.00

1.50

3.00

5.40 5.85 6.30 6.75 7.20

Run Number

Studentized Residuals

-3.00

-1.50

0.00

1.50

3.00

4 9 14 19 24 29 34 39
 A-26

Figure A-29. Mixture experiment: trace plot for RCT charge 

passed (natural log transform) 

Deviation from Reference Blend

28 day RCT

-0.238 -0.050 0.137 0.325 0.512

300

400

500

600

700

800

900

water

cement

silica fume

HRWRA

coarse agg

fine agg

Run Number

Cook's Distance

0.00

0.13

0.25

0.38

0.50

4 9 14 19 24 29 34 39

Figure A-28. Mixture experiment: Cook’s distance for RCT charge passed 

(natural log transform) 
 A-27

Figure A-31. Mixture experiment: contour Figure 52. Cook’s Distance for RCT (natural log transform) plot of ln (RCT charge passed) in 

water, silica fume, and HRWRA 

Figure A-30. Mixture experiment: contour plot of ln (RCT charge passed) in 

water, silica fume, and coarse aggregate 

A: water

0.184

C: silica fume

0.037

E: Coarse agg

0.424

0.400 0.013

0.160

5.8

6.0

6.2

6.4

6.6

6.8

A: water

0.175

C: silica fume

0.028

D: HRWRA

0.0198

0.0046 0.013

0.160

5.8

6.0

6.2

6.4

6.6
 A-28

Figure A-32. Mixture experiment: contour plot of ln (RCT charge passed) in 

cement, HRWRA, and fine aggregate 

B: cement

0.159

D: HRWRA

0.0333

F: Fine agg

0.279

0.250 0.0046

0.130

5.600

5.625

5.650

5.675
B-1

APPENDIX B. Experiment Design and Data Analysis for Factorial Experiment 

B.1 Experiment Design and Response Data 

Table B-1. Factorial experiment: design by volume fraction of factors 

Std 

Order 

Run 

Order Point FactorA w/c 

Factor B

Fine Agg

Factor C 

Coarse Agg

Factor D 

HRWRA 

Factor E 

Silica Fume 

 17 1 Center 0.39525 0.2712 0.4212 0.0060 0.0200 

 11 2 Fact 0.3576 0.2853 0.4071 0.0069 0.0247 

 7 3 Fact 0.3576 0.2853 0.4353 0.0051 0.0247 

 10 4 Fact 0.4329 0.2571 0.4071 0.0069 0.0247 

 4 5 Fact 0.4329 0.2853 0.4071 0.0051 0.0247 

 12 6 Fact 0.4329 0.2853 0.4071 0.0069 0.0153 

 2 7 Fact 0.4329 0.2571 0.4071 0.0051 0.0153 

 1 8 Fact 0.3576 0.2571 0.4071 0.0051 0.0247 

 18 9 Center 0.39525 0.2712 0.4212 0.0060 0.0200 

 8 10 Fact 0.4329 0.2853 0.4353 0.0051 0.0153 

 9 11 Fact 0.3576 0.2571 0.4071 0.0069 0.0153 

 6 12 Fact 0.4329 0.2571 0.4353 0.0051 0.0247 

 3 13 Fact 0.3576 0.2853 0.4071 0.0051 0.0153 

 14 14 Fact 0.4329 0.2571 0.4353 0.0069 0.0153 

 15 15 Fact 0.3576 0.2853 0.4353 0.0069 0.0153 

 13 16 Fact 0.3576 0.2571 0.4353 0.0069 0.0247 

 19 17 Center 0.39525 0.2712 0.4212 0.0060 0.0200 

 5 18 Fact 0.3576 0.2571 0.4353 0.0051 0.0153 

 16 19 Fact 0.4329 0.2853 0.4353 0.0069 0.0247 

 25 20 Axial 0.39525 0.2712 0.4494 0.0060 0.0200 

 21 21 Axial 0.47055 0.2712 0.4212 0.0060 0.0200 

 23 22 Axial 0.39525 0.2994 0.4212 0.0060 0.0200 

 27 23 Axial 0.39525 0.2712 0.4212 0.0078 0.0200 

 20 24 Axial 0.31995 0.2712 0.4212 0.0060 0.0200 

 31 25 Center 0.39525 0.2712 0.4212 0.0060 0.0200 

 26 26 Axial 0.39525 0.2712 0.4212 0.0042 0.0200 

 28 27 Axial 0.39525 0.2712 0.4212 0.0060 0.0106 

 24 28 Axial 0.39525 0.2712 0.3930 0.0060 0.0200 

 29 29 Axial 0.39525 0.2712 0.4212 0.0060 0.0294 

 22 30 Axial 0.39525 0.243 0.4212 0.0060 0.0200 

 30 31 Center 0.39525 0.2712 0.4212 0.0060 0.0200 
B-2

Table B-2. Factorial experiment: slump and 1-day strength data 

Run Point Slump (mm) 1-Day Strength (MPa) 

1 Center 76 70 16.6 16.0 16.2 

2 Fact 44 44 22.2 22.5 23.0 

3 Fact 13 13 22.5 17.9 22.0 

4 Fact 102 102 15.8 16.7 16.8 

5 Fact 57 57 16.4 16.7 16.1 

6 Fact 140 146 13.4 14.2 13.2 

7 Fact 70 64 13.9 11.1 13.7 

8 Fact 13 13 17.7 22.8 20.5 

9 Center 89 83 17.9 18.6 18.7 

10 Fact 102 102 15.2 15.2 15.2 

11 Fact 140 140 20.9 20.7 20.4 

12 Fact 32 32 13.8 18.8 18.9 

13 Fact 13 13 24.5 23.3 24.7 

14 Fact 76 76 17.3 17.2 17.2 

15 Fact 13 13 22.7 19.5 21.8 

16 Fact 13 13 20.8 20.6 21.4 

17 Center 51 64 19.4 18.9 18.1 

18 Fact 32 25 21.4 22.2 22.1 

19 Fact 38 32 16.0 16.1 16.3 

20 Axial 38 38 18.7 19.3 18.9 

21 Axial 121 114 14.5 14.9 14.0 

22 Axial 70 64 18.0 17.8 17.5 

23 Axial 64 64 19.3 20.0 20.4 

24 Axial 19 13 26.0 25.4 27.7 

25 Center 83 76 20.2 17.3 19.6 

26 Axial 64 64 18.9 19.5 18.9 

27 Axial 152 152 16.4 17.0 16.9 

28 Axial 95 95 20.1 20.8 19.7 

29 Axial 38 32 17.2 20.1 18.0 

30 Axial 102 102 17.7 17.0 17.7 

31 Center 76 76 16.8 19.6 18.8 
B-3

Table B-3. Factorial experiment: 28-day strength and RCT charge passed data 

Run Point 28-Day Strength (MPa) RCT Charge Passed (coulombs)

1 Center 54.0 63.0 – – 263 296 300 

2 Fact 59.4 60.2 – – 186 472 268 

3 Fact 53.5 52.8 51.7 – 151 151 178 

4 Fact 62.7 60.8 55.1 62.9 280 329 279 

5 Fact 52.6 56.6 55.8 – 273 236 262 

6 Fact 60.4 52.1 60.6 61.3 553 585 485 

7 Fact 50.1 49.4 51.5 – 468 550 490 

8 Fact 51.0 56.3 47.2 55.2 253 240 208 

9 Center 63.5 62.5 – – 247 315 305 

10 Fact 53.8 54.2 56.3 – 460 437 439 

11 Fact 61.6 64.5 60.8 – 415 427 393 

12 Fact 56.5 55.3 56.8 – 258 258 240 

13 Fact 58.1 50.2 54.4 – 343 362 317 

14 Fact 52.3 46.6 52.1 – 596 527 481 

15 Fact 60.3 58.7 58.6 – 218 288 330 

16 Fact 60.2 58.5 62.9 – 208 194 216 

17 Center 59.9 54.5 55.5 – 299 327 318 

18 Fact 58.4 60.1 56.3 – 360 364 340 

19 Fact 65.0 59.9 63.9 – 242 243 206 

20 Axial 56.7 57.0 62.1 – 168 242 224 

21 Axial 53.9 51.1 56.7 – 463 461 449 

22 Axial 60.7 62.9 63.6 – 319 305 257 

23 Axial 68.4 66.8 67.1 – 272 280 251 

24 Axial 62.1 59.4 54.0 – 190 184 192 

25 Center 58.2 55.8 56.8 – 239 246 287 

26 Axial 52.9 48.2 51.6 – 258 281 280 

27 Axial 51.3 55.2 56.8 – 704 766 644 

28 Axial 50.0 55.2 54.8 – 268 302 351 

29 Axial 55.5 53.8 56.2 – 163 170 153 

30 Axial 50.4 49.0 51.6 – 340 304 351 

31 Center 55.8 53.3 56.5 – 262 294 274 
B-4

B.2 Data Analysis and Model Fitting 

B.2.1 Slump 

Table B-4. Factorial experiment: sequential model sum of squares for slump 

Source Sum of Squares DF Mean Square F Value Prob > F 

Mean 130296.6 1 130296.6 – – 

Linear 31972.38 5 6394.48 9.10 < 0.0001 

2FI 10539.29 10 1053.93 2.25 0.0755 

Quadratic 2598.70 5 519.74 1.18 0.3856 

Cubic (aliased) 2316.86 5 463.37 1.10 0.4593 

Residual 2104.40 5 420.88 – – 

Total 179828.3 31 5800.91 – – 

Table B-5. Factorial experiment: lack-of-fit test for slump 

Source Sum of Squares DF Mean Square F Value Prob > F 

Linear 17103.61 21 814.46 7.15 0.0345 

2FI 6564.32 11 596.76 5.24 0.0618 

Quadratic 3965.62 6 660.94 5.80 0.0553 

Cubic (aliased) 1648.75 1 1648.75 14.47 0.0190 

Pure error 455.64 4 113.91 – – 

Table B-6. Factorial experiment: ANOVA for slump model 

Source Sum of Squares DF Mean Square F Value Prob > F 

Model 40109.97 7 5730.00 13.99 < 0.0001 

 A 12138.75 1 12138.75 29.63 < 0.0001 

 B 606.52 1 606.52 1.48 0.2360 

 C 6048.38 1 6048.38 14.77 0.0008 

 D 2426.07 1 2426.07 5.92 0.0231 

 E 10752.67 1 10752.67 26.25 < 0.0001 

 AB 1837.19 1 1837.19 4.48 0.0452 

 CD 6300.39 1 6300.39 15.38 0.0007 

Residual 9421.67 23 409.64 – – 

 Lack of fit 8966.03 19 471.90 4.14 0.0887 

 Pure error 455.64 4 113.91 – – 

Cor total 49531.64 30 – – – 
B-5

Table B-7. Factorial experiment: summary statistics for slump model 

Std. Dev. 20.24 R-Squared 0.8098 

Mean 64.83 Adj R-Squared 0.7519 

C.V. 31.22 Pred R-Squared 0.6275 

PRESS 18452.19 Adeq Precision 15.6200 

Table B-8. Factorial experiment: coefficient estimates for slump model 

Factor Coefficient

Estimate DF Standard Error

95% CI

Low

95% CI

High

Intercept 64.83 1 3.64 57.31 72.35 

A (w/c) 22.49 1 4.13 13.94 31.04 

B (fine agg) -5.03 1 4.13 -13.57 3.52 

C (coarse agg) -15.87 1 4.13 -24.42 -7.33 

D (HRWRA) 10.05 1 4.13 1.51 18.60 

E (silica fume) -21.17 1 4.13 -29.71 -12.62 

AB 10.72 1 5.06 0.25 21.18 

CD -19.84 1 5.06 -30.31 -9.38 

Model equation for slump in terms of coded factors: 

Slump = 64.83 + 22.49*A – 5.03*B – 15.87*C + 10.05*D – 21.17*E + 10.72*A*B – 

19.84*C*D 

Model equation for slump in terms of actual factors:

Slump = –1365.5 – 4876.9*w/c – 8334.7*fine agg + 8256.5*coarse agg 

 + 6.6982 x 105*HRWRA – 4503.6*silica fume + 20185*w/c*fine agg 

 – 1.564 x 106 *coarse agg*HRWRA 
B-6

Figure B-1. Factorial experiment: normal probability plot for slump 

Figure B-2. Factorial experiment: raw data plot for slump (hollow squares 

indicate control runs) 

Studentized Residuals

Normal % Probability

-2.77 -1.45 -0.13 1.18 2.50

1

5

10

20

30

50

70

80

90

95

99

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35

Run

Slump (mm)
B-7

Figure B-3. Factorial experiment: scatterplot of slump vs. w/c 

Figure B-4. Factorial experiment: scatterplot of slump vs. coarse aggregate 

R2 = 0.2451

0

20

40

60

80

100

120

140

160

180

0.3 0.35 0.4 0.45 0.5

w/c

Slump (mm)

R2 = 0.1221

0

20

40

60

80

100

120

140

160

180

0.38 0.4 0.42 0.44 0.46

CA (vol fraction)

Slump (mm)
B-8

Figure B-6. Factorial experiment: scatterplot of slump vs. HRWRA 

Figure B-5. Factorial experiment: scatterplot of slump vs. fine aggregate 

R2 = 0.0122

0

20

40

60

80

100

120

140

160

180

0.23 0.25 0.27 0.29 0.31

FA (vol fraction)

Slump (mm)

R2 = 0.049

0

20

40

60

80

100

120

140

160

180

0.003 0.005 0.007 0.009

HRWRA (vol fraction)

Slump (mm)
B-9

Figure B-7. Factorial experiment: scatterplot of slump vs. silica fume 

Figure B-8. Factorial experiment: means plots for slump 

R2 = 0.2171

0

20

40

60

80

100

120

140

160

180

0.01 0.015 0.02 0.025 0.03

silica fume (vol fraction)

Slump (mm)

0

20

40

60

80

100

120

140

160

180

Factor

Slump (mm)

w /c SF CA FA HRWRA
B-10

Figure B-10. Factorial experiment: lag plot for slump 

Figure B-9. Factorial experiment: slump vs. run sequence 

0

50

100

150

200

0 5 10 15 20 25 30 35

Run

Slump (mm)

0

50

100

150

200

0 50 100 150 200

lag slump (mm)

slump (mm)
B-11

B.2.2 1-Day Strength 

 

Table B-9. Factorial experiment: sequential model sum of squares for 1-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F 

Mean 10780.54 1 10780.54 – – 

Linear 215.97 5 43.19 22.15 < 0.0001 

2FI 27.35 10 2.73 1.92 0.1236 

Quadratic 13.60 5 2.72 3.48 0.0441 

Cubic (aliased) 1.86 5 0.37 0.31 0.8865 

Residual 5.95 5 1.19 – – 

Total 11045.26 31 356.30 – – 

Table B-10. Factorial experiment: lack-of-fit test for 1-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F 

Linear 43.98 21 2.09 1.75 0.3129 

2FI 16.63 11 1.51 1.27 0.4444 

Quadratic 3.03 6 0.51 0.42 0.8339 

Cubic (aliased) 1.18 1 1.18 0.98 0.3772 

Pure error 4.78 4 1.19 – – 

Table B-11. Factorial experiment: ANOVA for 1-day strength model 

Source Sum of Squares DF Mean Square F Value Prob > F 

Model 240.87 8 30.11 27.76 < 0.0001 

 A 213.26 1 213.26 196.66 < 0.0001 

 B 0.48 1 0.48 0.45 0.5113 

C 0.043 1 0.043 0.040 0.8433 

E 2.06 1 2.06 1.90 0.1819 

A

2 6.20 1 6.20 5.72 0.0257 

AC 5.15 1 5.15 4.75 0.0404 

AE 7.16 1 7.16 6.60 0.0175 

BC 6.51 1 6.51 6.00 0.0227 

Residual 23.86 22 1.08 – – 

Lack of fit 19.08 18 1.06 0.89 0.6248 

Pure error 4.78 4 1.19 – – 

Cor total 264.72 30 – – – 
B-12

Table B-12. Factorial experiment: summary statistics for 1-day strength model 

Std. Dev. 1.04 R-Squared 0.9099 

Mean 18.65 Adj R-Squared 0.8771 

C.V. 5.58 Pred R-Squared 0.8294 

PRESS 45.16 Adeq Precision 22.583 

Table B-13. Factorial experiment: coefficient estimates for 1-day strength model 

Factor Coefficient

Estimate DF Standard Error

95% CI

Low

95% CI

High

Intercept 18.29 1 0.24 17.80 18.79 

A (w/c) -2.98 1 0.21 -3.42 -2.54 

B (fine agg) 0.14 1 0.21 -0.30 0.58 

C (coarse agg) 0.043 1 0.21 -0.40 0.48 

E (silica fume) 0.29 1 0.21 -0.15 0.73 

A2 0.46 1 0.19 0.061 0.86 

AC 0.57 1 0.26 0.027 1.11 

AE 0.67 1 0.26 0.13 1.21 

BC -0.64 1 0.26 -1.18 -0.098 

Model equation for 1-day strength in terms of coded factors:

1-day strength = 18.29 – 2.98*A + 0.14*B + 0.043*C + 0.29*E + 0.46*A2 + 0.57*A*C 

 + 0.67*A*E – 0.64*B*C 

Model equation for 1-day strength in terms of actual factors: 

1-day strength = –63.8 – 860.8*w/c + 1361.3*fine agg + 450.8*coarse agg – 1431.5*silica fume 

 +323.9*(w/c)2 + 1068*w/c*coarse agg + 3780*w/c*silica fume 

 – 3208*fine agg*coarse agg 
B-13

Figure B-11. Factorial experiment: normal probability plot for 1-day strength

Figure B-12. Factorial experiment: raw data plot for 1-day strength (hollow 

squares indicate control runs) 

Studentized Residuals

Normal % Probability

-1.97 -0.94 0.09 1.13 2.16

1

5

10

20

30

50

70

80

90

95

99

5

10

15

20

25

30

0 5 10 15 20 25 30

Run

1-day strength (MPa)
B-14

Figure B-13. Factorial experiment: scatterplot of 1-day strength vs. w/c 

Figure B-14. Factorial experiment: scatterplot of 1-day strength vs. fine aggregate

R2 = 0.8056

10

15

20

25

30

0.3 0.35 0.4 0.45 0.5

w/c

1-day strength (MPa)

R2 = 0.0018

10

15

20

25

30

0.23 0.25 0.27 0.29 0.31

FA (vol fraction)

1-day strength (MPa)
B-15

Figure B-15. Factorial experiment: scatterplot of 1-day strength vs. coarse aggregate

Figure B-16. Factorial experiment: scatterplot of 1-day strength vs. HRWRA 

R2 = 0.0002

10

15

20

25

30

0.38 0.4 0.42 0.44 0.46

CA (vol fraction)

1-day strength (MPa)

R2 = 0.0004

10

15

20

25

30

0.003 0.005 0.007 0.009

HRWRA (vol fraction)

1-day strength (MPa)
B-16

Figure B-17. Factorial experiment: scatterplot of 1-day strength vs. silica fume 

Figure B-18. Factorial experiment: means plot for 1-day strength 

R2 = 0.0078

10

15

20

25

30

0.01 0.015 0.02 0.025 0.03

silica fume (vol fraction)

1-day strength (MPa)

10

15

20

25

30

Factor

1-day strength (MPa)

w /c SF CA FA HRWRA
B-17

Figure B-19. Factorial experiment: 1-day strength vs. run sequence 

Figure B-20. Factorial experiment: lag plot for 1-day strength 

10

15

20

25

30

0 5 10 15 20 25 30 35

Run

Strength (MPa)

10

15

20

25

30

10 15 20 25 30

lag 1-day strength (MPa)

Strength (MPa)
B-18

B.2.3. 28-Day Strength 

Table B-14. Factorial experiment: sequential model sum of squares for 28-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F

Mean 100300.0 1 100300.0 – – 

Linear 292.28 5 58.46 5.71 0.0012 

2FI 129.24 10 12.92 1.53 0.2208 

Quadratic 27.53 5 5.51 0.56 0.7318 

Cubic (aliased) 61.25 5 12.25 1.62 0.3050 

Residual 37.84 5 7.57 – – 

Total 100848.1 31 3253.17 – – 

Table B-15. Factorial experiment: lack-of-fit test for 28-day strength 

Source Sum of Squares DF Mean Square F Value Prob > F 

Linear 219.79 21 10.47 1.16 0.4972 

2FI 90.55 11 8.23 0.91 0.5942 

Quadratic 63.02 6 10.50 1.16 0.4620 

Cubic (aliased) 1.76 1 1.76 0.20 0.6813 

Pure error 36.08 4 9.02 – – 

Table B-16. Factorial experiment: ANOVA for 28-day strength model 

 

Source Sum of Squares DF Mean Square F Value Prob > F

Model 300.21 4 75.05 7.87 0.0003 

 A 16.57 1 16.57 1.74 0.1990 

 D 223.41 1 223.41 23.43 < 0.0001 

 E 7.47 1 7.47 0.78 0.3842 

 AE 52.76 1 52.76 5.53 0.0265 

Residual 247.94 26 9.54 – – 

 Lack of fit 211.86 22 9.63 1.07 0.5389 

 Pure error 36.08 4 9.02 – – 

Cor total 548.15 30 – – – 
B-19

Table B-17. Factorial experiment: summary statistics for 28-day strength model 

Std. Dev. 3.09 R-Squared 0.5477 

Mean 56.88 Adj R-Squared 0.4781 

C.V. 5.43 Pred R-Squared 0.3997 

PRESS 329.05 Adeq Precision 9.964 

Table B-18. Factorial experiment: coefficient estimates for 28-day strength model 

Factor Coefficient

Estimate DF Standard Error

95% CI

Low

95% CI

High

Intercept 56.88 1 0.55 55.74 58.02 

A (w/c) -0.83 1 0.63 -2.13 0.46 

D (HRWRA) 3.05 1 0.63 1.76 4.35 

E (silica fume) 0.56 1 0.63 -0.74 1.85 

AE 1.82 1 0.77 0.23 3.40 

Model equation for 28-day strength in terms of coded factors: 

28-day strength = 56.88 – 0.83*A + 3.05*D + 0.56*E + 1.82*A*E 

Model equation for 28-day strength in terms of actual factors: 

28-day strength = 124.0 – 227.3*w/c + 3390*HRWRA – 3937.5*silica fume 

+ 10262*w/c*silica fume 
B-20

Figure B-21. Factorial experiment: normal probability plot for 28-day strength

Figure B-22. Factorial experiment: raw data plot for 28-day strength (hollow 

squares indicate control runs) 

Studentized Residuals

Normal % Probability

-2.34 -1.25 -0.17 0.92 2.01

1

5

10

20

30

50

70

80

90

95

99

40

45

50

55

60

65

70

0 5 10 15 20 25 30 35

Run

28-day strength (MPa)
B-21

Figure B-23. Factorial experiment: scatterplot of 28-day strength vs. w/c 

Figure B-24. Factorial experiment: scatterplot of 28-day strength vs. fine aggregate

R2 = 0.0302

45

50

55

60

65

70

0.3 0.35 0.4 0.45 0.5

w/c

28-day strength (MPa)

R2 = 0.0703

45

50

55

60

65

70

0.23 0.25 0.27 0.29 0.31

FA (vol fraction)

28-day strength (MPa)
B-22

Figure B-26. Factorial experiment: scatterplot of 28-day strength vs. HRWRA 

Figure B-25. Factorial experiment: scatterplot of 28-day strength vs. coarse aggregate 

R2 = 0.0115

45

50

55

60

65

70

0.38 0.4 0.42 0.44 0.46

CA (vol fraction)

28-day strength (MPa)

R2 = 0.4076

45

50

55

60

65

70

0.003 0.005 0.007 0.009

HRWRA (vol fraction)

28-day strength (MPa)
B-23

Figure B-27. Factorial experiment: scatterplot of 28-day strength vs. silica fume 

FIgure B-28. Factorial experiment: means plot for 28-day strength 

R2 = 0.0136

45

50

55

60

65

70

0.01 0.015 0.02 0.025 0.03

silica fume (vol fraction)

Strength (MPa)

45

50

55

60

65

70

Factor

28-day strength (MPa)

w /c SF CA FA HRWRA
B-24

Figure B-30. Factorial experiment: lag plot for 28-day strength 

Figure B-29. Factorial experiment: 28-day strength vs. run sequence 

40

50

60

70

80

0 5 10 15 20 25 30 35

Run

Strength (MPa)

45

50

55

60

65

70

45 50 55 60 65 70

lag 28-day strength (MPa)

Strength (MPa)
B-25

B.2.4 RCT Charge Passed (coulombs) 

Table B-19. Factorial experiment: sequential model sum of squares for RCT charge 

 passed 

Source Sum of Squares DF Mean Square F Value Prob > F 

Mean 3155867 1 3155867 – – 

Linear 393517.2 5 78703.43 27.43 < 0.0001 

2FI 15156.50 10 1515.65 0.40 0.9252 

Quadratic 44410.76 5 8882.15 7.31 0.0040 

Cubic (aliased) 10046.83 5 2009.37 4.77 0.0557 

Residual 2104.61 5 420.92 – – 

Total 3621103 31 116809.8 – – 

Table B-20. Factorial experiment: lack-of-fit test for RCT charge passed 

Source Sum of Squares DF Mean Square F Value Prob > F 

Linear 69614.70 21 3314.99 6.30 0.0432 

2FI 54458.20 11 4950.75 9.41 0.0221 

Quadratic 10047.44 6 1674.57 3.18 0.1410 

Cubic (aliased) 0.61 1 0.61 .001 0.9745 

Pure error 2104.00 4 526.00 – – 

Table B-21. Factorial experiment: ANOVA for RCT charge passed model 

 

Source Sum of Squares DF Mean Square F Value Prob > F 

Model 441455.3 6 73575.89 74.25 < 0.0001 

 A 81666.67 1 81666.67 82.42 < 0.0001 

 B 6868.17 1 6868.17 6.93 0.0146 

 C 11440.67 1 11440.67 11.55 0.0024 

 E 292604.2 1 292604.2 295.30 < 0.0001 

 E

2 38369.41 1 38369.41 38.72 < 0.0001 

 AE 10506.25 1 10506.25 10.60 0.0034 

Residual 23780.54 24 990.86 – – 

 Lack of fit 21676.54 20 1083.83 2.06 0.2537 

 Pure error 2104.00 4 526.00 – – 

Cor total 465235.9 30 – – – 
B-26

Table B-22. Factorial experiment: summary statistics for RCT charge passed model

Std. Dev. 31.48 R-Squared 0.9489 

Mean 319.06 Adj R-Squared 0.9361 

C.V. 9.87 Pred R-Squared 0.8784 

PRESS 56577.00 Adeq Precision 34.166 

Table B-23. Factorial experiment: coefficient estimates for RCT charge passed model 

Factor Coefficient

Estimate DF Standard Error

95% CI

Low

95% CI

High

Intercept 291.11 1 7.22 276.20 306.01 

 A (w/c) 58.33 1 6.43 45.07 71.59 

 B (fine agg) -16.92 1 6.43 -30.18 -3.66 

 C (coarse agg) -21.83 1 6.43 -35.09 -8.57 

 E (silica fume) -110.42 1 6.43 -123.68 -97.16 

 E2 36.11 1 5.80 24.14 48.09 

 AE -25.63 1 7.87 -41.87 -9.38 

Model equation for RCT charge passed in terms of coded factors: 

RCT charge passed = 291.11 + 58.33*A – 16.92*B – 21.83*C – 110.42*E + 36.11*E2 

– 25.63*A*E 

Model equation for RCT charge passed in terms of actual factors: 

RCT charge passed = 635.4 + 4445.6*w/c – 1199.8*fine agg – 1548.5*coarse agg 

– 31651*silica fume + 1.635 x 106*(silica fume)2 

– 1.448 x 105*w/c*silica fume 
B-27

Figure B-31. Factorial experiment: normal probability plot for RCT charge passed 

Figure B-32. Factorial experiment: raw data plot for RCT charge passed (hollow squares

indicate control runs) 

Studentized Residuals

Normal % Probability

-2.50 -1.30 -0.10 1.11 2.31

1

5

10

20

30

50

70

80

90

95

99

0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25 30 35

Run

RCT charge passed (coulombs)
B-28

Figure B-33. Factorial experiment: scatterplot of RCT charge passed vs. w/c 

Figure B-34. Factorial experiment: scatterplot of RCT charge passed vs. fine aggregate 

R2 = 0.1755

0

100

200

300

400

500

600

700

800

0.3 0.35 0.4 0.45 0.5

w/c

Charge passed (coulombs)

R2 = 0.0148

0

100

200

300

400

500

600

700

800

0.23 0.25 0.27 0.29 0.31

FA (vol fraction)

Charge passed (coulombs)
B-29

 

Figure B-35. Factorial experiment: scatterplot of RCT charge passed vs. coarse aggregate

Figure B-36. Factorial experiment: scatterplot of RCT charge passed vs. HRWRA 

R2 = 0.0246

0

100

200

300

400

500

600

700

800

0.38 0.4 0.42 0.44 0.46

CA (vol fraction)

Charge passed (coulombs)

R2 = 0.002

0

100

200

300

400

500

600

700

800

0.003 0.005 0.007 0.009

HRWRA (vol fraction)

Charge passed (coulombs)
B-30

 

Figure B-37. Factorial experiment: scatterplot of RCT charge passed vs. silica fume 

Figure B-38. Factorial experiment: means plot for RCT charge passed 

R2 = 0.6289

0

100

200

300

400

500

600

700

800

0.01 0.015 0.02 0.025 0.03

silica fume (vol fraction)

Charge passed (coulombs)

0

100

200

300

400

500

600

700

800

Factor

Charge Passed (coulombs)

w /c SF CA FA HRWRA
B-31

 

Figure B-40. Factorial experiment: lag plot for RCT charge passed 

Figure B-39. Factorial experiment: RCT charge passed vs. run sequence 

0

200

400

600

800

0 5 10 15 20 25 30 35

Run

Charge passed (coulombs)

0

100

200

300

400

500

600

700

800

0 200 400 600 800

lag charge passed (coulombs)

Charge passed (coulombs)
C-1

APPENDIX C. COST User’s Guide 

USER’S GUIDE 

July, 2001 

Marcia J. Simon 

Dale P. Bentz 

James J. Filliben
C-2

ABSTRACT 

This user’s guide provides instructions for and examples of using the Concrete Optimization 

Software Tool (COST), a joint product of the Federal Highway Administration and the National 

Institute of Standards and Technology. COST provides an Internet-based system for optimizing 

concrete performance based on statistical experiment design and analysis methods. Working 

with local raw materials, COST designs an experimental program of concrete mixtures to be 

prepared and evaluated. In these mixtures, the user can vary the water-to-cement (w/c) ratio and 

other concrete mixture parameters such as the cement, mineral and chemical admixture, and 

aggregate contents. Once the measured responses (properties) for the prepared concretes are 

input into the COST system, it analyzes the results and determines the optimum mixture 

proportions based on user-supplied performance criteria. Results and analysis are provided in 

both graphical and numerical formats to aid in interpretation. Typical uses of COST might be to 

design a concrete that meets all specifications at minimum cost or to design a concrete that 

provides maximum durability within a specific cost range. 

Keywords: Building technology, concrete, experiment design, mixture proportioning, 

optimization, response surfaces.
C-3

SECTION 1 

Overview 

C1.1 Introduction 

In the simplest case, portland cement concrete is a four-component mixture of water, portland 

cement, fine aggregate, and coarse aggregate. Additional components, such as chemical 

admixtures (air entraining agents, superplasticizers) and mineral admixtures (coal fly ash, silica 

fume, blast furnace slag), may be added to the basic mixture to enhance certain properties of the 

fresh or hardened concrete. High-performance concrete mixtures, which may be required to 

meet several performance criteria (e.g., compressive strength, elastic moduli, rapid chloride 

permeability) simultaneously, typically contain at least six components. Thus, optimizing 

mixture proportions for high-performance concrete, which contains many constituents and is 

often subject to several performance constraints, can be a difficult and time-consuming task. 

The Concrete Optimization Software Tool (COST) is an online interactive system developed to 

assist engineers, concrete producers, and researchers in optimizing portland cement concrete 

mixtures for their particular applications. COST applies response surface methodology (RSM), a 

collection of statistical experiment design and analysis methods, to the problem of optimizing 

concrete mixture proportions. RSM is often used in industry for product development, 

formulation, and improvement, and is applicable to problems such as concrete mixture 

proportioning where several input variables (factors) influence a performance measure 

(response). 

COST is intended to provide an introduction to concrete practitioners who are unfamiliar with 

the concepts and process of applying RSM to concrete mixture proportioning. COST allows 

users to learn how RSM can be applied to the problem of optimizing concrete mixtures. 

There are two scenarios for which COST could be applied: 

1. The first, and probably most common, would be the case where a user wants to proportion a 

concrete mixture to meet a set of specifications at minimum material cost. 

2. The second is the case where the user wants to maximize (or minimize) a particular response 

or responses, irrespective of cost. 

COST can be used to optimize cement paste, mortar, or concrete mixtures. In all three cases, 

varying the mixture component proportions affects both fresh and hardened properties of the 

paste, mortar or concrete. The properties (responses) depend on the proportions of the 

components. Table C-1 lists examples of typical components and responses for concrete 

mixtures (components and responses other than those listed can be used). 

 

In COST, the water-cement (w/c) ratio (or water-cementitious materials (w/cm) ratio) is varied 

along with up to four additional components. These are referred to as variable factors. Other 

factors may be included in the mixture at fixed (constant) levels, and are referred to as fixed 

factors. Up to five concrete properties, or responses, (e.g., slump, strength, air content, cost, etc.) 
C-4

can be designated by the user according to the requirements of the application. These concepts 

are explained more fully in section 2. 

Table C-1. Examples of components and responses 

Components Responses

Water 

Cement (including blended cements) 

Mineral admixtures 

(e.g., fly ash, silica fume, slag, 

metakaolin) 

Chemical admixtures 

(water reducers, retarders, air 

entraining agents) 

Aggregate 

Fresh properties 

(e.g., slump, air content, unit 

weight, temperature, set time) 

Mechanical properties 

(e.g., strength, modulus of elasticity, 

shrinkage, creep) 

Durability 

(e.g., freeze-thaw, scaling, alkali 

silica reaction, sulfate attack, 

abrasion) 

COST is accessible via the Internet. The program consists of a front-end HTML interface that 

allows the user to enter required information. Underlying code (written in C) processes the 

input, generates the experiment designs and mixture proportions, calls routines for statistical 

analysis, and generates output. The statistical analysis routines are part of an interactive 

statistical software package, DATAPLOT, which was developed at the National Institute of 

Standards and Technology (NIST). 

C1.2 Scope 

COST is not intended to supplant or compete with commercially available experiment design 

and analysis software packages. Rather, the purpose of COST is to introduce to the concrete 

practitioner the concepts of statistical experiment design and analysis using RSM and how they 

might be applied to concrete mixture proportioning. COST is specifically geared toward the 

application of these methods to concrete mixture proportioning. 

This section provides a general overview of the COST program. Section 2 of this manual 

provides step-by-step instructions for using COST. Section 3 contains a glossary of terms, 

additional details on the statistical aspects of the experiment designs and analyses used, and a list 

of references. 

C1.3 System Requirements 

To use COST, your system must have the following components and settings: 

• Personal computer (Pentium®1 or equivalent) with video card and monitor set for 800 x 600 

resolution (min) and 65536 colors. 

 

1

 Certain commercial products are identified to completely specify the COST system. In no case does such 

identification imply endorsement by NIST or the FHWA or that the identified products are the best available for 

the purpose. 
C-5

• Access to the Internet (World Wide Web) and one of the following browsers: Netscape 

Navigator 4.0 (or above) or Microsoft® Internet Explorer 4.0 (or above). The COST 

interface uses both frames and JavaScript. 

C1.4 Disclaimer 

This software was developed at NIST by employees of the Federal Government in the course of 

their official duties. Pursuant to Title 17, Section 105 of the U.S. Code, this software is not 

subject to copyright protection and is in the public domain. COST is an experimental system. 

NIST and the Federal Highway Administration (FHWA) assume no responsibility whatsoever 

for its use by other parties, and make no guarantees, expressed or implied, about its quality, 

reliability, or any other characteristic. We would appreciate acknowledgement if the software is 

used. 

The U.S. Department of Commerce and the U.S. Department of Transpotation make no 

warranty, expressed or implied, to users of COST and associated computer programs, and 

accepts no responsibility for its use. Users of COST assume sole responsibility under Federal 

law for determining the appropriateness of its use in any particular application; for any 

conclusions drawn from the results of its use; and for any actions taken or not taken as a result of 

analyses performed using these tools. 

Users are warned that COST is intended for use only by those competent in the field of concrete 

technology and is intended to supplement the informed judgment of the qualified user. Lack of 

accurate predictions by the COST models could lead to erroneous conclusions with regard to 

materials selection and design. All results should be evaluated by an informed user. 

C1.5 General Information 

C1.5.1 COST Homepage and Main Menu 

The COST homepage may be accessed at http://ciks.cbt.nist.gov/cost. The COST homepage is 

shown in figure C-1. 

The homepage contains a brief overview of the COST program. The blue bar on the left side of 

the screen is the main menu. Menu selections are described briefly below. Details on each step 

are provided in section 2, “Using COST.” 
C-6

Figure C-1. COST homepage 

COST Home—allows user to return to the initial screen (figure C-1) at any time. 

Specify Responses—user enters information on responses to be measured. Responses are the 

measured properties of the concrete (fresh or hardened) such as slump, air content, strength, 

shrinkage, etc., that are specified for the particular project. 

Specify Mixtures—user enters information on mixture components and proportions, and COST 

generates an experimental plan. 

View Experimental Plan—user can view a previously generated experimental plan. 

Run Trial Batches—contains guidelines for performing the trial batches according to the 

experimental plan. This step is performed by the user in his/her laboratory, and involves 

batching, fabricating, curing, and testing specimens. Types of specimens and test methods used 

will depend on the responses specified in step 1. It is the user’s responsibility to determine the 

appropriate test method to use. 

Enter Results—user enters test results for each trial batch and each response. 

Analyze Data—COST provides 10 analysis tasks to assist the user in analyzing and interpreting 

the results. The tasks include checking the experiment design, looking at trends in the data 

graphically, generating empirical (quadratic) models for each response, and optimizing according 

to individual runs, means, and models. 

Summarize Analysis—summary of main results of analysis. 

Menu 

Frame 

Links to 

NIST & 

FHWA 

homepages 

Overview 

of COST 
C-7

User’s Guide—HTML version of the COST User’s Guide. A PDF version is also available for 

download. 

FAQ’s—frequently asked questions. 

References—a list of references on statistics, response surface methods and DATAPLOT 

(software used by COST). 

Glossary—a glossary of statistical terms. 
C-8
C-9

SECTION 2 

Using COST 

C2.1 Background and Preliminary Planning 

Using COST requires several steps, including planning, running trial batches, entering results, 

analyzing, interpreting, and summarizing results. These tasks have been divided into the 

following six steps (as listed in the COST menu): 

• Specify responses. 

• Specify mixtures. 

• Run trial batches. 

• Enter results. 

• Analyze data. 

• Summarize analysis. 

In most cases, these steps will be performed in the order listed above. Each step is described in 

detail in the sections below. 

Before starting the six-step process, the user must perform some preliminary steps: 

1. Define the overall objective of the project. Typical objectives include the following: 

o Minimize cost while meeting several performance criteria for responses. 

o Minimize or maximize a single response or several responses. 

2. Define the properties (responses) and mixture components (factors) to be included, and 

define which will be variable or fixed factors. Variable factors include w/c or w/cm plus up 

to four additional components. Additional factors may be included at fixed levels. See 

“Background Information” below for further details. 

3. Define the performance criteria (most likely based on the job specifications) for each 

response, and the numerical ranges for each factor. 

4. Collect necessary material information (e.g., properties and costs of each component). The 

required information for each type of material is listed in table C-2. 

C2.1.1 Responses 

Responses are the concrete properties of interest that will be measured and compared to specified 

performance criteria (i.e., limits on allowable values of the responses). The responses are 

dependent variables; that is, the value of a measured response depends on the settings of the 

independent variables, or factors (see step 2 below). Responses and performance criteria are 

often dictated by specifications. For example, the specifications for a particular job may indicate 

that the concrete must have a slump between 50 mm and 100 mm, an air content between 4.5 

percent and 7.5 percent by volume, and 28-day strength greater than 69 MPa. The responses in 
C-10

this case are slump, air content and 28-day strength, and the performance criteria are the ranges 

of acceptable values of the responses. 

C2.1.2 Factors 

The factors are the independent variables that affect the measured values of the responses. For 

concrete mixtures, these factors include mixture proportions (relative amounts of each 

component material) as well as others related to construction practice and environmental 

conditions. COST assumes that construction and environmental conditions are fixed (as in the 

case of a set of laboratory or plant trial batches), so the factors of concern are the mixture 

proportions for each component. 

Concrete can contain a variety of component materials. Allowable material types for this 

version of COST include the following: 

• Water. 

• Cement. 

• Mineral admixtures (up to 4): fly ash, silica fume, slag, other (user specified). 

• Chemical admixtures (up to 3): all user specified. 

• Aggregates (up to 3): coarse, fine, other (user specified). 

COST always requires that either w/c or w/cm be included as a variable factor. Thus, the two 

mixture components water and cement are accounted for in this single factor. 

Factors may be variable or fixed (set at a constant level). For concrete mixture proportioning, 

variable factors would usually be the mixture components expected to have the most significant 

effects on the responses. Fixed factors would be those expected to have little or no effect, and 

would be held constant in the experiment. Any of the factors included in COST may be set as 

variable or fixed; however, COST limits the user to a maximum of five variable factors for any 

one experiment (the greater the number of variable factors, the greater the number of trial 

batches required). Because w/c or w/cm is always considered to be one factor, up to six material 

components (water, cement, and four others) may be varied. 

For all variable factors, low and high settings, or levels, must be defined. The low and high 

settings are the range over which the factor will vary. For example, w/c could have a range of 

0.35 (low level) to 0.45 (high level), or silica fume could have a range of 5 to 10 percent (cement 

mass replacement). For fixed factors, a fixed (constant) level is specified. 

Table C-2 summarizes the information required for different types of materials. 

 

Once the user has decided on the factors to include, defined their ranges (for variable factors) or 

constant levels (for fixed factors), and obtained other necessary information (table C-2), the 

information may be entered into COST to generate a trial batch plan. 

In most cases, the selection of components and their ranges is up to the user; however, in some 

cases, some of the factors and levels may be designated in specifications. For example, a 
C-11

specification may have a maximum w/c, or minimum silica fume content. COST does not 

provide guidance on the selection of minimum and maximum values for the components. 

Table C-2. Information required for different materials 

Material Information Required

Water None 

Cement Specific gravity 

Cost ($/kg) 

Mineral 

admixture 

Replacement rate (percent mass fraction of cement) 

Specific gravity 

Cost ($/kg) 

Chemical 

admixture 

Dosage rate (liters per kg cement) 

Specific gravity 

Percent solids (by mass fraction) 

Cost ($/liter) 

Aggregates 

Volume fraction (or mass fraction) 

Specific gravity 

Absorption (%) 

Moisture content (%) 

Cost ($/kg) 
C-12

C2.2 Step 1—Specify Responses 

When “Specify Responses” is selected from the main menu, a form entitled “COST Input Form: 

Response Information” appears in the right frame. This form is shown in figure C-2. 

Figure C-2. “Response Information” form 

Referring to figure C-2, the following information must be entered into COST for each response: 

• A name for the response. The name should be as short as possible (no more than 10 

characters) and may include alphanumeric characters and underscores (e.g., slump, 

28-day_str). No other characters are allowed. 

• The units in which the response is measured. 

• A lower limit for the response, if applicable. This would be a specified minimum value 

(e.g., minimum 1-day strength of 15 MPa). If there is not a lower limit for the response, a 

default value of –100 is used. 

• An upper limit for the response, if applicable. This would be a specified maximum value 

(e.g., maximum rapid chloride permeability (RCT) test result of 1000 coulombs). If there is 

no upper limit, a default value of 99999 is used. 
C-13

• A result weight factor between 0 and 1, to indicate the relative importance of the response 

in optimization, compared to the other responses. A factor of 1 indicates most important. 

The default value is 1 for all responses. If all responses are of equal importance, use the 

default values. (Note: to optimize a single response while ignoring all others, set the weight 

factor of the response of interest to 1 and the weight factors of all others to zero).

• The type of weight function to use in optimization. Choices are: 

o Minimum value is best (linear decreasing from 1 to 0 over response range). 

o Maximum value is best (linear increasing 0 to 1 over response range). 

o Target value is best (linear increasing from 0 to 1 over lower half of response range, and 

decreasing from 1 to 0 over upper half of response range). 

o Within range (all values in range are equally acceptable but no values outside range are 

acceptable). 

The response range is defined by the lower and upper limits specified above, or by the 

minimum (or maximum) value of the response obtained in the experiment if it is greater than 

(less than) the lower (upper) limit. A different type of weight function can be specified for 

each response, if desired. 

• A filename for the project, eight characters or fewer (alphanumeric characters only) and no 

extension. The filename will be unique to a particular project, and COST will create several 

files using the same filename with different extensions as you proceed through the steps 

(Note: remember this filename, as it will be needed in subsequent steps as well). 

After entering the filename, the user should click on “Submit” to submit the completed form 

information to COST, or “Reset” to reset all settings to their default values. 
C-14

C2.3 Step 2—Specify Mixtures 

When “Specify Mixtures” is selected from the main menu, a form entitled “COST Input Form: 

Mixture Factors and Information” appears. Figure C-3 shows the first two sections of this form. 

The instructions for completing these sections are listed below. 

Figure C-3. First two sections of “Mixture Factors and Information” input form 

C2.3.1 Instructions for Section 1: Number of Parameters (Factors) to Vary 

In section 1, the number of parameters (factors) to vary is selected. The user may select 2 to 5 

parameters to vary (default is 4). The number of experimental runs is also shown for each 

selection. The number of experimental runs depends on whether the user includes 3 or 5 center 

points in the design (the number of center points is entered at the bottom of the form). 

• To select the number of parameters to vary, use the pulldown menu. 

C2.3.2 Instructions for Section 2: Select w/c or w/cm 

In section 2, the user selects w/c or w/cm as a factor, defines the range (low and high settings) 

for this factor, and enters information about the cement. 

• To select w/c or w/cm , click on the radio button next to the desired choice. 

• Enter the low and high settings of w/c or w/cm (by mass fraction) in the boxes labeled 

“min” and “max”, respectively. 

• Enter the cement specific gravity.

• Enter the cement cost in dollars per kilogram. 
C-15

C2.3.3 Instructions for Section 3: Select Other Mixture Components 

The third section of the form allows selection of other mixture components (mineral admixtures, 

chemical admixtures, and aggregates). A maximum of four additional variable factors, and any 

number of fixed factors, may be included. The total number of variable factors selected 

(including w/c or w/cm, selected in section 2) must match the “number of parameters to vary” 

selected in section 1. 

The additional factors are selected from three types of materials: mineral admixtures, chemical 

admixtures, and aggregates. Regardless of type, the first task is to indicate whether the factor 

will be included or not, and if it will be included, whether it will be variable or fixed. This 

setting is defined using a pulldown menu on the left of the factor name, as described in the 

following instructions (refer to figure C-4 below): 

• To include a component as a variable factor, select “On” in the pulldown menu at the left. 

Enter the type of material (if not predefined), the low and high settings in the “min” and 

“max” boxes, and the additional information in the other boxes. 

• To include a component as a fixed factor (held at a constant level), select “Off” in the 

pulldown menu on the left, and enter a nonzero fixed setting in the “min” box (the entry in 

the “max” box will be ignored). Then enter the additional information for each fixed factor 

in the boxes. 

• To exclude a component completely, select “Off” in the pulldown menu on the left, and set 

the value in the “min” box to zero. All other information for these factors is ignored. NOTE: 

this is the default setting for all factors. 

Figure C-4. Third section of “Mixture Factors...” form (mineral admixtures section) 
C-16

C2.3.4 Specific Instructions for Mineral Admixtures 

There are three pre-designated mineral admixtures (fly ash, silica fume, and slag), plus one blank 

for a user-designated choice. Mineral admixtures ranges are defined in terms of percent cement 

mass replacement. Specific gravity and cost must also be entered. 

• Enter the name of the mineral admixture (if the user-defined box is used). 

• To define the range for a mineral admixture, enter the min and max values in units of 

percent cement mass replacement. 

• Enter the specific gravity of the mineral admixture. 

• Enter the cost of the mineral admixture in dollars per kilogram. 

C2.3.5 Specific Instructions for Chemical Admixtures 

All chemical admixtures are user-designated, and chemical admixtures ranges are defined in 

terms of dosage rate in liters per kg of cement. 

Figure C-5. Third section of “Mixture Factors...” input form (chemical admixtures 

 section) 

• Enter the name of the chemical admixture. 

• To define the range for a chemical admixture, enter the min and max values in units of 

dosage in liters per kilogram of cement (note that this is L/kg, not L/100 kg!!). 

• Enter the specific gravity of the chemical admixture. 

• Enter the percent solids (by mass fraction) of the chemical admixture. 
C-17

• Enter the cost of the chemical admixture in dollars per liter. 

C2.3.6 Specific Instructions for Aggregates 

Aggregates include two predesignated types (coarse and fine), and one blank for a userdesignated choice. The user-designated choice may be used for an additional aggregate or fibers. 

Aggregates may be defined in terms of volume fraction or mass fraction. Steps for entering 

aggregate information (see figure C-6) are as follows: 

Figure C-6. Third section of “Mixture Factors...” form (aggregates section) 

• Select how the aggregate range will be defined (by volume or by mass) using the pulldown 

menu at the top of the section (see figure 6). 

• Enter a name for the aggregate (if the user-defined box is used). 

• To define the range for the aggregate, enter the min and max values in units of volume 

fraction or mass fraction (depending on the selection made above). 

 

• Enter the bulk specific gravity for the aggregate. 

• Enter the absorption of the aggregate (in percent by mass of aggregate). 

• Enter the moisture content of the aggregate as batched (in percent by mass of aggregate). 

• Enter the cost of the aggregate in dollars per kilogram. 

C2.3.7 Instructions for Section 4: Additional Information 

In section 4, the user enters additional information needed to generate the trial batch 

experimental plan. This information includes the number of center points to be run and a 

random number seed. Steps for entering this information (see figure C-7) are as follows: 
C-18

Figure C-7. Fourth section of “Mixture Factors...” form (additional information section)

• Enter the number of center points to use in the experiment, using the pulldown menu. 

Center points are experimental runs in which all factors are set at the midpoints of their 

ranges. The coded values for the low and high settings are –1 and +1, respectively. The 

coded values for the center points are zero. Center point mixes are replicated to estimate 

pure error and also may be used as control mixes to assess variation over time. It is a good 

idea to run at least 3 center points to assess this variation. 

• Enter a random number seed (any number less than zero). This is used to generate a 

random run order for the experiment. 

• Enter the filename in which to store the information. This name should be the same as that 

entered in “Specify Responses.” Again, the filename is entered without an extension (COST 

adds the appropriate extensions as necessary). 

After entering the filename, the user may click on “Submit” to submit the information to COST 

and generate the experimental plan, or the user may click “Reset” to set all values back to their 

defaults (all information entered will be lost). 

When “Submit” is clicked, COST processes the input and generates an experimental plan, which 

is displayed on the screen. The user should print this plan using the “PRINT” command in the 

browser. An example of a portion of an experimental plan generated by COST is shown in 

figure C-8. 
C-19

Figure C-8. Portion of an experimental plan generated by COST

The columns in the printed table shown above correspond to the run number (the order in which 

the mixtures should be prepared), the mixture number (the mixture number according to standard 

experiment design tables), mixture proportions in terms of the mass of each component per cubic 

meter of concrete, and an estimated cost for each mixture based on the individual material costs 

provided by the user. 

If the plan is not printed immediately, it can be viewed and printed at a later time by selecting 

“View Expt Plan” from the main menu. 
C-20

C2.4 Step 3—Run Trial Batches 

The next step after generating an experimental plan is to actually perform the experiment. The 

experiment in this case is a set of trial batches from which specimens will be fabricated and 

tested for the responses and mixture components specified in steps 1 and 2. Before running the 

experiment, steps 1 and 2 must be complete, and the user should have a printed copy of the 

experimental plan containing the mixture proportions for the set of trial batches. 

When “Run Trial Batches” is selected from the main menu, the screen shown in figure C-9 

appears. This screen does not require any input; rather, it provides instructions and guidelines 

for running the trial batches. These guidelines are also provided below. 

Figure C-9. “Run Trial Batches” screen 

C2.4.1 Guidelines for Running Trial Batches 

Running the experiment is the most time-consuming task because it involves physically running 

the experimental plan and collecting data on the performance variables of interest. Running the 

experiment includes the following tasks: 

• Ordering and preparing the materials needed for the experiment. 

• Ensuring that equipment is available and ready for use. 

• Coordinating personnel. 

• Batching, fabricating, testing, and recording data (running the experiment). 
C-21

The following sections describe important considerations to keep in mind while performing the 

experiment. 

C2.4.2 Nuisance Factors and Run Sequence Randomization 

There are three types of factors that affect the responses in an experiment: 

• Variable factors. 

• Fixed factors. 

• Nuisance factors. 

The primary goal of the experiment is to determine optimal settings of the variable factors 

specified in step 1. These are the major factors of interest. Depending on the objective, some 

fixed factors may have also been designated. These are less important factors that are held 

constant throughout the experiment. 

The variable factors and fixed factors are controlled in the experiment. However, in addition to 

these, there are other factors that are not controlled in the experiment but which could possibly 

affect the experimental results. These are called nuisance factors. It is often assumed that these 

nuisance factors do not, or should not, have any effect, but in reality they may have an effect. 

Nuisance factors may include the following: 

• Mid-experiment changes in instruments, equipment, environmental conditions, (temperature, 

pressure, humidity, etc.), measuring devices. 

• Test procedures/protocols. 

• Day of week. 

• Time of day. 

• Operators. 

Nuisance factors may affect the measured test results, which would in turn affect the data 

analysis, and ultimately the final conclusions (i.e., the estimated values for the optimal mixture 

proportions). 

Run sequence randomization is used to minimize the effect of nuisance factors. Experiment 

designs are usually generated in a “standard order” based on the settings of the factors. This 

order (used by COST in the data analysis) is indicated by the “mixture number” column (column 

2) in the experimental plan generated by COST (figure C-8). Run sequence randomization is the 

general experiment design technique in which random numbers are assigned to each of the 

specified runs in the experiment, and these random numbers determine the order in which the 

experiment is to be run (the “run order” or “run sequence”). The experimental plan generated by 

COST is printed in run order—the first column of the experimental plan, labeled “run number,” 

is the run order to be used for the experiment (figure C-8). 

It is very important to follow the run sequence in order to minimize possible error in the 

experimental results caused by known or unknown nuisance factors in the experiment. 
C-22

C2.4.3 Running the Experiment 

The quality and accuracy of the final mixture proportion settings will depend very much on the 

care taken in carrying out the experiment. The following is a list of recommended practices: 

• Attention to detail, consistency and proper execution in batching, mixing, fabricating, curing, 

testing, and recording results are essential. 

• For each trial batch, use several (preferably 3 or more) specimens for each test. 

• The experimental plan contains 3 or 5 center point runs. These are replicates (repeats of 

batches using the same settings) which can be used during the experiment as control mixes. 

Scheduling to allow for one control mixture per week is recommended—in this way 

significant week-to-week variation can be detected. 

• If possible, the same operator should perform the same tasks throughout the experiment. 

• Mistakes will inevitably occur even in the best laboratories, and it is important to 

acknowledge this and to be prepared to repeat a batch if it is suspected that an error has 

occurred. 
C-23

C2.5 Step 4—Input Results 

This section describes how to input experimental results into the COST program for analysis. 

The following instructions assume that steps 1, 2, and 3 have been successfully completed. 

Within this step, you may perform the following tasks: 

• Change cost information (optional). 

• review the experimental plan (optional). 

• enter or edit the experimental results (data) for analysis (required). 

Instructions for each of these tasks are provided below. 

C2.5.1 Instructions for Changing Cost Information 

Before entering the test results, you may optionally change the costs of the raw materials, 

if you have updated information since the project began. Please note that these updates 

should be made before entering in the data (test results) as described below; otherwise, 

the new costs will not be in effect during the analysis phase. To do so, click on the link “Update 

cost information.” You will see a box with the caption “Datafile name”. Enter the name of your 

datafile (no extension) in the box, and press the “Submit” button. A form entitled “COST Input 

Form: Update Material Costs for ‘DATAFILE’” will appear. Make any necessary changes to the 

material costs and press “Submit” to save the changes. 

If you make a mistake and would like to reset all costs to their default values, press “Reset,” 

before pressing the “Submit” button. If you decide not to change the costs, simply press the back 

button on your browser to return to the previous page, or select any entry from the blue menu 

sidebar. 

C2.5.2 Instructions for Entering/Editing Data 

When you are ready to enter your data, click on “Enter or edit data in chronological (run) 

order.” You will then see a screen entitled “COST Input Form: Testing Results (Project 

Filename).” Enter the project filename (with no extension) and press “Submit.” 

The next screen will be “COST Input Form: Test Results for ‘DATAFILE.’” The first part of this 

form, as shown in figure C-10, allows you to make changes to the response information that you 

entered in step 1 (instructions are the same as for step 1 described previously). If you do not 

have any changes to make in the response information, simply scroll down to the second part of 

the form. 

The second part of the form is for entry (or editing) of the experimental results, or data. The last 

two lines in the figure C-10 show two rows of entries for the experimental test results. The first 

column shows the run order for the experiment, and the second column shows the mixture 

number (these should correspond to the run and mixture numbers in your experimental plan). 

The third column is “Cost” (the first response). 
C-24

Figure C-10. Data entry form for the COST system 

IMPORTANT: Because the total cost of each mixture is calculated from the individual materials 

costs entered in step 1, the third column contains nonzero entries that should not be changed. 

The responses requiring data entry start in the fourth column (one column per response). 

• If you are entering results for the first time, these columns will contain zeros. Enter the 

appropriate results in these columns for each mixture. 

• If you are editing a file you already created, these columns will contain nonzero values. 

Perform any necessary editing. 

IMPORTANT: As you enter or edit your data, please check the input as you go for typographical 

errors and to make sure that the results are entered in the correct columns and rows. 

It should be noted that some of the input values shown in figure C-10 are outside of the 

acceptable range specified by the lower and upper limits, as will most likely be the case for any 

real experiment. 

When data entry is complete, you may use the “PRINT” button on your browser to print the 

input form. 

IMPORTANT: It is highly recommended that you check the printed form for accuracy, and edit 

the information if necessary (to edit, simply follow the instructions in this section). 
C-25

C2.6 Step 5—Analyze Data 

 

The next step after entering the data is to analyze the results, with the ultimate goal of 

determining optimal mixture proportions. The analysis techniques employed by COST consist 

of both graphical analysis and numerical analysis (modeling). The analysis is broken down into 

10 tasks, which are described in detail below. 

C2.6.1 Instructions for Changing Response Information 

Before analyzing the data, COST gives the user the option of changing response variable limits, 

weights, and function types. To do so, click on “Change response variable limits, weights, and 

function types.” You will then be prompted for the filename. After entering the filename, press 

“Submit” and follow the instructions in the section “Step 1—Specify Responses.” 

C2.6.2 Analysis Tasks 

The analysis tasks are listed by the purpose of the task followed by the statistical tool used to 

perform the task, as shown in figure C-11. For example, the purpose of task 2 is to “Assess the 

Balance of the Design,” and the tool used is “Counts Plot Matrix of Factors.” An example and 

explanation of each task and tool is provided below. The output of each task is a GIF file that is 

generated by DATAPLOT. The GIF file contains tabular output, graphical output, or a 

combination of both. 

Figure C-11. Analysis menu showing individual analysis tasks 
C-26

C2.6.2.1 Task 1: Characterize the Response Variables 

This task provides a quantitative summary of the data for each response. The result of this task 

is a table of descriptive statistics such as mean, range, and standard deviation for each response. 

An example is provided below (figure C-12). 

Figure C-12. Summary statistics table (output of analysis task 1)

The summary statistics provided for each response and the total score (TS) are described in table 

C-3 (next page). 
C-27

Table C-3. Description of summary statistics provided in analysis task 1 

Statistic Description 

PROJECT GOAL optimization goal for response 

SPEC MIN minimum value specified by user 

SPEC MAX maximum value specified by user 

DATA COUNT number of data points read(runs) 

DATA # IN SPEC number of runs with response meeting spec 

DATA % IN SPEC percentage of runs with response meeting spec 

DATA MIN minimum response value 

DATA MEAN mean response value 

DATA MEDIAN median response value 

DATA MAX maximum response value 

DATA RANGE range of response values (max – min) 

DATA SD sample standard deviation 

DATA REL. SD SSD relative to mean (coefficient of variation) 
C-28

C2.6.2.2 Task 2: Assess the Balance of the Design 

This task is a check to make sure that the design is balanced. The result of this task is a plot 

similar to figure C-13. Figure C-13 shows a matrix of plots showing the number of design points 

(experimental runs) at each setting for all combinations of two factors. For example, the 

highlighted box in figure C-13 shows the number of design points for the factors X2 and X3 

(fine aggregate and coarse aggregate). There are nine numbers in this box, representing 

different settings of the factors. The lower left number indicates that there are 4 design points 

that have the setting “-1, -1” (in coded values) for fine aggregate and coarse aggregate. For this 

design every set of two factors has the same experimental layout (the sets of nine numbers in 

each box are the same). For any design generated by COST, this will always be the case. The 

percentage in the upper left corner indicates the correlation coefficient. Ideally, this will be zero. 

For any design generated by COST, the correlation coefficient will be zero for all sets of factors, 

indicating a balanced design. 

Figure C-13. Output of analysis task 2 (counts plot matrix of factors) 
C-29

C2.6.2.3 Task 3: Assess Optimality of Design Points for All Responses Jointly 

The purpose of task 3 is to see how all the responses compare to the specifications for each 

design point. Figure C-14 shows the output, which is a matrix of plots comparing each 

combination of two factors. In each large box there are nine smaller boxes, corresponding to the 

nine possible design settings for two factors. In each smaller box there will be between one and 

five numbers, depending on the number of responses being investigated. In the example below, 

there are five responses, and thus five numbers. The legend at the lower left of the plot indicates 

which number in the small box corresponds to which response (Y1, Y2, Y3, Y4, Y5). In the 

example below, the large box for X2 and X3 is highlighted, and the small box corresponding to 

settings of X2 = 0, X3 = 0 is highlighted. The numbers in the small box indicate that for these 

settings of X2 and X3, there was 1 response for Y1 that was within the acceptable region, there 

were 7 for Y2, 2 for Y3, 11 for Y4, and 11 for Y5. This information allows the user to assess 

how well the ranges selected for the factors allow him to meet the desired specifications. For Y1 

and Y3, only a few responses met the specification. Therefore, for this set of mixture 

proportions, it may be difficult to optimize these responses. A new set of acceptance criteria 

could be defined, or different ranges for the mixture proportion factors may be needed. Other 

analysis tasks will give the user a sense of which direction the factors must be shifted. 

Figure C-14. Output of analysis task 3 (counts plot matrix of factors)
C-30

C2.6.2.4 Task 4: Assess Optimality of Design Points for All Responses Jointly 

Task 4 is similar to task 3 in assessing optimality of design points. For each combination of two 

variable settings (e.g., X2 = 0, X3 = 0), the percentage of runs falling in at least one admissible 

region (for all responses taken together) is shown (see figure C-15). The overall percentage for 

all the runs is given in the first text line below the title. In this case, it is 59 percent. The overall 

percentage for the number of design points meeting all “n” (in this case, 5) acceptance criteria 

for responses is indicated on the second line below the title. In this case, it is zero. The gray 

squares over the numbers in the boxes indicate the highest percentage in each box, and the 

triangles indicate the lowest percentage in the box. This gives a quick visual cue to the settings 

that are best in meeting the acceptance criteria. 

Figure C-15. Output of analysis task 4 
C-31

C2.6.2.5 Task 5A: Determine Interrelationships between Response Variables 

Figure C-16 shows a matrix of scatterplots showing data for each combination of two response 

variables (RVs). The admissible region for each pair of RVs is indicated as a gray box 

surrounded by dashed lines. These plots give a sense of relationships between responses and 

also a sense of how many points fall in the admissible region (as defined by the performance 

criteria set by the user) for each pair of responses. The numbers in the upper left corner of each 

box (e.g., 2/31 = 6 percent) indicate the number of responses falling in the admissible region. 

The large gray shaded box at the bottom left of the entire plot shows the relative ease or 

difficulty of meeting the performance criteria (i.e., falling within the admissible region) for 

single responses and for pairs of responses. 

Figure C-16. Output of analysis task 5A 
C-32

C2.6.2.6 Task 5B: Interrelationships between Response Variables and Factors 

The output for this task (shown in figure C-17) shows the relationship between responses and 

factors. In each plot, the response values (Y axis) are shown for each factor level (X axis). The 

correlation coefficient (indicating the strength of the linear relationship between Y and X) is 

shown in the upper left corner of each plot. The stronger the linear relationship, the closer this 

value will be to 1 or –1 (depending on the slope). Examining these plots allows the user to 

assess which factors are important (controlling) for each response. The gray shaded boxes at the 

bottom of the plot summarize the control factors (left box, percentage indicates correlation 

coefficient) and the “weak” factors (right box). 

.

Figure C-17. Output of analysis task 5B 
C-33

C2.6.2.7 Task 6: Assess Relationship between Response Variables and Factors 

This task provides an assessment of the relationship between responses and factors by examining 

plots of the mean (average) values of the responses at each factor level. Figure C-18 shows the 

output for this task. For each response, there is a plot of the mean values at each level of each 

factor. The gray shaded boxes indicate the admissible region for each response. Influential 

factors are those that have a definite slope (for example, silica fume for response Y1, cost, or w/c 

ratio for Y3, 1-day strength). The steeper the slope, the more important the factor. A flat line (or 

nearly flat) indicates little effect of the factor (for example, fine aggregate for Y5, RCT). 

.

Figure C-18. Output of analysis task 6
C-34

C2.6.2.8 Task 7A: Best Settings for Each Factor Based on Means 

This task provides a graphical means of selecting best factor settings based on mean (average) 

values of “scoring functions” calculated for each response separately as well as a total score 

function (TS). The total score is a weighted linear combination of scores calculated for each 

response (the weight given to each response is defined in “Step 1—Specify Responses,” as the 

result weight, a value ranging from zero to 1). Figure C-19 shows the output from this task.

The best settings (as coded values) are shown in parentheses on the right side of each plot. The 

best setting based on total score are shown in the gray box at the bottom of the plot, in both 

coded and actual units. As in task 6, a steep slope indicates a large influence of a particular 

factor on the score, while a flat slope indicates little or no influence. 

Figure C-19. Output of analysis task 7A 
C-35

C2.6.2.9 Task 7B: Best Factors Based on Individual Runs 

This task produces a plot (figure C-20) showing the best settings for total score (weighted linear 

combination of scores for individual responses) based on individual runs. The total scores for 

each run are calculated, sorted (lowest to highest) and plotted along the X axis. The Y axis is 

used to differentiate between the individual runs. Each run is indicated by its number in the 

experimental plan, and the coded values of each factor are provided in parentheses on the right. 

These values are staggered for readability. The topmost number has the highest total score. 

. 

Figure C-20. Output of analysis task 7B 
C-36

C2.6.2.10 Task 8: Model Fitting and Verifications 

In this task, an empirical mathematical model is fitted to the data for each response, and to the 

total score, using the standard regression methods (least squares). A full quadratic model is fit 

initially, and then reduced by eliminating terms with significance level less than 0.05. When the 

task is selected, the response to be fitted is selected using a pull-down menu. In addition to the 

response variables (up to five), a model may also be fitted to total score. For a complete analysis 

of all responses, task 8 must be thus executed multiple times, once for total score and once for 

each response variable. Output similar to figure C-21 is produced for the each selected response 

variable. The output provides a plot for the response as a function of each factor, showing all 

response data for the coded values of each factor. These plots may indicate trends (see for 

example, the plot for silica fume in figure C-21, which indicates a downward trend in RCT test 

results with increasing silica fume). In addition to the plots, a summary box of the important 

terms in the model is provided to the right of the second row of plots, and the actual model is 

printed below the plots. The model is in terms of CODED values (the model can be translated to 

actual factor values). The model can be used to predict the response values for settings other 

than those used in the experiment (but within the experimental space)—a calculator for doing 

this is provided in task 10. 

Figure C-21. Output of analysis task 8 (for response RCT) 
C-37

C2.6.2.11 Task 9: Numerical Optimization 

This task uses numerical optimization techniques to identify the optimal settings for total score, 

over the entire experimental space. This optimization is performed during the model fitting for 

total cost in task 8, so when task 9 is executed, the COST system simply returns a table 

indicating the best settings as determined by the numerical optimization, as shown in figure C22. 

Figure C-22. Output of analysis task 9 
C-38

C2.6.2.12 Task 10: Response Prediction 

As shown in figure C-23, this task provides a calculator for predicting response values using the 

models from task 8. The user enters values for each factor (in terms of actual values) and the 

program calculates the responses and the total score. Calculations can be performed for up to 10 

different combinations of factors. 

Figure C-23. Calculator for predicting responses based on models 
C-39

C2.7 Step 6—Summarize Analysis 

This step simply returns a table summarizing the three different optimum settings (from analysis 

tasks 7A, 7B, and 9) determined by the COST system, as shown in figure C-24. 

Figure C-24. Example summary returned by the COST system
C-40
C-41

SECTION 3 

References 

1. Box, G.E.P., Hunter, W.G., and J.S. Hunter, Statistics for Experimenters. New York, John 

Wiley & Sons, 1978. 

2. Myers, R.H. and D.C. Montgomery, Response Surface Methodology: Process and Product 

Optimization Using Designed Experiments. New York, John Wiley & Sons, 1995. 

3. Simon, M.J., Lagergren, E.S, and L.G. Wathne, “Optimizing High-Performance Concrete 

Mixtures Using Statistical Response Surface Methods.” In Proceedings of the 5th

International Symposium on Utilization of High-Strength/High-Performance Concrete. 

Norwegian Concrete Association, Oslo, Norway, June, 1999, pp. 1311-1321. 

4. Heckert, A., and J.J. Filliben, DATAPLOT Reference Manual Volume I: Commands, 

National Institute of Standards and Technology, 1999.