Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home   |   Pavements Home
Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-RD-03-060

Concrete Mixture Optimization Using Statistical Methods

Previous | Table of Contents

APPENDIX C. Cost User's Guide

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.

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:

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.) 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.

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:

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 Transportation 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."

Figure C-1. COST homepage. Picture. This figure shows a screenshot of the COST homepage, indicating the links to FHWA and NIST homepages on the upper left, the menu frame for selecting each of the six steps in using COST on the left side, and the main window, which contains an overview of COST.
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.

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.

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):

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:
    • Minimize cost while meeting several performance criteria for responses.
    • 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 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:

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 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
WaterNone
CementSpecific gravity
Cost ($/kg)
Mineral admixtureReplacement rate (percent mass fraction of cement)
Specific gravity
Cost ($/kg)
Chemical admixtureDosage rate (liters per kg cement)
Specific gravity
Percent solids (by mass fraction)
Cost ($/liter)
AggregatesVolume fraction (or mass fraction)
Specific gravity Absorption (%)
Moisture content (%)
Cost ($/kg)

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. Picture. This figure shows the form for inputting response information. The form is a matrix in which the column headings are the chosen responses (up to five), and the row headings (on the left) are response name, units, lower limit, upper limit, result weight (0 to 1), and weight function (these are described in the text of the user's guide). All of the input boxes are text input except for weight function, which is a drop-down menu. Underneath the form is another input box for a file name to store the selected inputs. Detailed instructions for filling in this section are provided in the text of the user's guide.
Figure C-2. "Response Information" form

Referring to figure C-2, the following information must be entered into COST for each response:

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.

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. Picture. This figure shows the first two sections of the form for inputting mixture factors and information. The first section contains a pull-down menu from which the user selects the number of parameters (factors) to vary. The second section contains one row with two radio buttons on the left and four input boxes to the right of them. The radio buttons allow the user to select either water-cement ratio or water-cementitious materials ratio as a factor. The input boxes are labeled “min,” “max,” “cement specific gravity,” and “cement cost (dollars per kilogram).” Detailed instructions for filling in this section are provided in the text of the user's guide.
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).

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.

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):

Figure C-4. Third section of "Mixture Factors..." form (mineral admixtures section). Picture. This figure contains the first part of the third section of the mixture factors input form, in which information about other mixture components is entered. Mineral admixture information is entered in this section. There are four rows, three labeled “silica fume,” “fly ash,” “slag,” and one with a user definable label. On the left of each label is an “on/off” drop-down menu. To the right of each label are four input boxes, labeled (across the top) “min,” “max,” “specific gravity,” and “cost (dollars per kilogram).’ The user inputs the required information for each admixture to be included. Detailed instructions for filling in this section are provided in the text of the user's guide.
Figure C-4. Third section of "Mixture Factors..." form (mineral admixtures section)

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.

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). Picture. This figure shows a continuation of the third section of the mixture factors input form, in which chemical admixture information is entered. This section has three rows and six columns of text input boxes. The columns are labeled “Type,” “min,” “max,” “specific gravity,” “percent solids by mass,” and “cost (dollars per liter).” On the left of the “Type” column is an on/off selection button for each row. Detailed instructions for filling in this section are provided in the text of the user's guide.
Figure C-5. Third section of "Mixture Factors..." input form (chemical admixtures section)

C2.3.6 Specific Instructions for Aggregates

Aggregates include two predesignated types (coarse and fine), and one blank for a user-designated 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). Picture. This figure shows a further continuation of the third section of the mixture factors input form, in which aggregate information is entered. At the top of this section is a drop-down menu to select volume or mass as the basis for defining aggregate factor ranges. Underneath this is a table with three rows and seven columns for user input. The column headings are “Type,” “min,” “max,” “Bulk specific gravity,” “absorption (percent)," "moisture content (percent),” and “cost (dollars per kilogram)”. Under “Type,” the first row is labeled “fine agg,” the second is labeled “coarse agg,” and the third is blank for user input. To the left of the “Type” column are “on/off” pull-down menus. Detailed instructions for filling in this section are provided in the text of the user's guide.
Figure C-6. Third section of "Mixture Factors..." form (aggregates section)

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:

Figure C-7. Fourth section of “Mixture Factors...” form (additional information section). Picture. This figure shows the fourth and final section of the mixture factors information form. Additional information needed to generate an experimental plan is entered here. The first item is a drop-down button to select the number of center points to run. Under this is an input box to enter a random number seed less than zero. Under this is another input box for entry of a filename to save the information. This filename should be the same as that used for the response information. Underneath the filename box are a “Submit” button and a “Reset to defaults” button.
Figure C-7. Fourth section of "Mixture Factors..." form (additional information section)

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.

Figure C-8. Portion of an experimental plan generated by COST. Picture. This figure shows a portion of a sample experimental plan generated by COST. The number of columns will vary depending on the number of parameters, but for this example there are nine columns, labeled “Run number,” “Mix number,” “water,” “cement,” “Coarse aggregate,” “Fine aggregate,” “Silica fume,” “HRWRA,” and “Cost.” The number of rows equals the number of mixes to be made. This example, a partial table, shows 18 rows. Each row contains the run order of the mix, the mix number according to standard experiment design format, mixture proportions in terms of mass per cubic meter of concrete, and estimated cost of each mix.
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.

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. Picture. This figure shows the beginning of the instructions and guidelines for running experiments, which are shown in the main window of COST when the user selects step 3, “Run Trial Batches.”
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:

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:

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:

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.

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:

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:

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).

Figure C-10. Data entry form for the COST system. Picture. This figure shows the top portion of a sample data entry form for the COST system. The top of this form is identical to the response input form shown in figure C-1. The top section provides a summary of the responses and settings the user has chosen. The second section has up to seven columns, depending on the number of responses. The first two columns are labeled “Run number” and “Mix number,” corresponding to the experimental plan. The next five columns are input boxes for entry of measured responses for each mix. The responses are entered under the appropriate "Response" heading at the top of the first section. Detailed instructions for data entry are provided in the user's guide.
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).

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).

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. Picture. This figure shows the analysis menu that is shown in the main window of COST when the user selects step 5, “Analyze Data.” Ten tasks are listed, each with a statistical tool underneath it. The statistical tool labels are links to each analysis step.
Figure C-11. Analysis menu showing individual analysis tasks

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).

TASK 1: SUMMARIZE THE 5 RESPONSE VARIABLES
DATA FILE = MJS98E
STAT TOOL: SUMMARY STATISTICS
Figure C-12. Summary statistics table (output of analysis task 1). Picture. This figure shows an example of output from analysis task 1. The output is a table with 13 rows and up to six columns containing summary statistics for the experiment. The column headings are labeled with each individual response and a total score. The rows are labeled “project goal,” “spec min,” “spec max,” “data count,” ”data number in spec,” “data percentage in spec,“ ”data min,” “data mean,” “data median,” “data max,” “data range,” “data SD,” and “data rel. SD.” These row headings are defined in table 3 of the user's guide.
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).

Table C-3. Description of summary statistics provided in analysis task 1
Statistic Description
PROJECT GOALoptimization goal for response
SPEC MINminimum value specified by user
SPEC MAXmaximum value specified by user
DATA COUNTnumber of data points read(runs)
DATA # IN SPECnumber of runs with response meeting spec
DATA % IN SPECpercentage of runs with response meeting spec
DATA MINminimum response value
DATA MEANmean response value
DATA MEDIANmedian response value
DATA MAXmaximum response value
DATA RANGErange of response values (max - min)
DATA SDsample standard deviation
DATA REL. SDSSD relative to mean (coefficient of variation)

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.

TASK 2: ASSESS THE QUALITY OF THE EXPERIMENT DESIGN MJS98R
TOTAL NUMBER OF DESIGN POINTS (RUNS) = 31
STAT TOOL: COUNTS PLOT MATRIX OF FACTORS
CHARACTER = NUMBER OF DESIGN POINT RUNS
LEGEND = CORR. COEFF. (BEST CC = 0% WORST CC = +/- 100%)
Figure C-13. Output of analysis task 2 (counts plot matrix of factors). Picture. This figure shows an example of output from analysis task 2. The output is a matrix of plots arranged like the upper portion of a matrix. The names of each factor form the diagonal of the matrix (X1 at top left, X5 at bottom right). There is a plot for each combination of two factors. Each plot contains a 3 by 3 matrix of numbers and a percentage (the correlation coefficient) in the upper left corner. The plot for X2 and X3 is highlighted in this example because it is referred to in the explanation of this set of plots that is found in the user's guide.
Figure C-13. Output of analysis task 2 (counts plot matrix of factors)

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.

TASK 3: ASSESS OPTIMALITY OF DESIGN POINTS FOR ALL 5 RESPONSES JOINTLY MJ S98R
% OF 5 X 31 "POINTS" IN SOME ADMINS. REGION - 91/155 = 59%
% of 31 DESIGN POINT RUNS IN ALL ADMIS REGIONS = 0/31 = 0%
CHARACTER = # OF RUNS IN ADMISSABLE REGION FOR EACH OF THE 5 RESPONSES
Figure C-14. Output of analysis task 3 (counts plot matrix of factors). Picture. This figure shows an example of output from analysis task 3. The output is a matrix of plots arranged like the upper portion of the matrix, with factor headings on the diagonal (X1 at top left, X5 at bottom right). Each plot consists of a box containing nine smaller boxes arranged in a 3 by 3 matrix within the larger box. There is a plot (large box) for each combination of two factors. The small boxes contain 5 numbers arranged in an “X” within the box (one number in each corner plus one in the middle). On the lower left of this output screen there is a separate box that is a legend for the small boxes. The meaning of the numbers is described in the text of the user's guide.
Figure C-14. Output of analysis task 3 (counts plot matrix of factors)

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.

TASK 4: ASSESS OPTIMALITY OF DESIGN POINTS FOR ALL 5 RESPONSES JOINTLY MJS98R
% OF 5 X 31 "POINTS" IN SOME ADMIS. REGIONS = 91/155 = 59%
% OF 31 DESIGN POINT RUNS IN ALL ADMIS. REGIONS = 0/31 = 0%
CHARACTER = % OF RUNS (ACROSS ALL 5 RESPONSES) FALLING IN ADMIS REGION
Figure C-15. Output of analysis task 4. Picture. This figure shows example output from analysis task 4. The output is a 5 by 5 matrix of plots, with factor headings on the diagonal (X1 at upper left, X5 at lower right). Each plot contains a 3 by 3 matrix of numbers. For each combination of two factor settings (total of 3 squared equals 9 settings), the numbers in the plot show the percentage of runs for which at least one response meets the performance criteria (falls in the admissible region). The lower left number corresponds to factor settings of negative 1, negative 1; the upper right corresponds to settings of 1, 1, and so forth. Some numbers are enclosed in boxes, indicating that they are the highest value in a plot, and some numbers are enclosed in downward pointing triangles, indicating that they are the lowest value in a plot.
Figure C-15. Output of analysis task 4

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.

TASK 5(A): DETERMINE INTERRELATIONSHIPS BETWEEN RESPONSE VARIABLES
STAT TOOL: SCATTER PLOT MATRIX OF RESPONSE VARIABLES
LEGEND = % OF THE 31 VALUES FALLING IN THE ADMISSABLE REGION
Figure C-16. Output of analysis task 5A. Picture. This figure shows example output from analysis task 5A. The output is an upper half matrix of plots with responses on the diagonal (Y1 on the upper left, Y5 on the lower right). The plots are scatterplots for each combination of two responses. Each scatterplot shows an admissible region, which is defined by the user's performance criteria (upper and lower limits) for the responses. Each box in the matrix, including the headings, also contains a line of text at the top with a fraction (for example, three thirty-firsts) and its equivalent in percent (for example, 10 percent). This indicates the number and percentage of runs in which results fall in the admissible region. At the bottom left of the output screen is a summary table showing the responses in order from easiest to hardest to meet for both single responses and pairs of responses. The easiest response or pair of responses has the highest percentage of runs falling in the admissible region.
Figure C-16. Output of analysis task 5A

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).

TASK 5(B): ASSESS RELATIONSHIP BETWEEN RESPONSE VARIABLES & FACTORS
STAT TOOL: SCATTER PLOT MATRIX OF RESPONSE VARIABLES VERSUS FACTORS
PLOT LEGEND = CORRELATION COEFFICIENT (TO MEASURE LINEAR RELATIONSHIP)
Figure C-17. Output of analysis task 5B. Picture. This figure shows an example output from task 5B. The output is a scatterplot matrix of responses versus factors. The rows represent responses (with headings on the left, Y1 at the top and Y5 at the bottom), and the columns represent factors (with headings at the bottom, X1 on the left, to X5 on the right). To the right of each row heading are scatterplots of that response and each factor. Each plot shows the data points, the admissible response region in gray, and a correlation coefficient in the upper left corner. On the far right side, after the last scatterplot in each row, is a scale for the plots in the units selected by the user. Underneath the scatterplot matrix are two shaded boxes containing summary tables. On the left is a summary table indicating the control (strongest) factor for each response and the corresponding correlation coefficient. On the right is a summary table showing weak factors for each response.
Figure C-17. Output of analysis task 5B

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).

TASK 6: ASSESS RELATIONSHIP BETWEEN RESPONSE VARIABLES & FACTORS
STAT TOOL: MEAN PLOTS OF RESPONSE VARIABLES VERSUS FACTORS
Figure C-18. Output of analysis task 6. Picture. This figure shows example output for analysis task 6. The output is a set of means plots in five rows. The first column is the response (Y1 at the top, Y5 at the bottom). The second column contains the means plots in order X1 through X5. The admissible region (with its upper and lower values) is shown in gray, and the overall mean is also shown as a horizontal line with its value. On the far right of each row is the scale in the units selected by the user.
Figure C-18. Output of analysis task 6

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.

TASK 7 (A): BEST FACTOR SETTINGS BASED ON MEAN VALUES
STAT TOOL: MEAN PLOTS OF LINEAR SCORE FUNCTION
(BEST SCORE = 1 WORST SCORE = 0)
Figure C-19. Output of analysis task 7A. Picture. This figure shows an example of output for analysis task 7A. The output is a set of means plots of the score function for each response and the total score (combination of all responses). There are six rows. On the left of each row is a heading with the response name in the center, “case equals” on the lower left, and “WT equals” on the lower right. The second column contains the means plots in order X1 through X5. In the top left corner of each means plot are the factor settings (in volume fraction), which give the best score. On the bottom right of each means plot are the factor settings (in coded values) for the best score. The scale for each plot is on the far right. Underneath the plots is a shaded box containing the best settings based on mean total score in coded and actual values.
Figure C-19. Output of analysis task 7A

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.

TASK 7 (B): BEST FACTOR SETTINGS BASED ON INDIVIDUAL RUNS
STAT TOOL: RANKED SCORES PLOT AND LIST
(BEST SCORE = 1 WORST SCORE = 0) PLOT CHARACTER = RUN SEQUENCE ID
Figure C-20. Output of analysis task 7B. Picture. This figure shows an example of output for analysis task 7B. The output is a plot of total score on the X-axis and run number on the Y-axis. Y-axis values are plotted from lowest to highest based on lowest to highest total score. To the right of this plot, within the plot frame, are the coordinates (in coded values) for each run. At the bottom of the screen in a shaded box are best settings based on individual run in both coded and actual values.
Figure C-20. Output of analysis task 7B

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.

RESPONSE VARIABLE 5: RCT
Figure C-21. Output of analysis task 8 (for response RCT). Picture. This figure shows an example of output from analysis task 8, the model fitting tool. The output shown here is for one of the responses, RCT. The output consists of plots of the response, RCT (on the Y-axis) versus each factor (on the X-axis). There is a separate plot for each factor. On the bottom right is a box in which important model terms are listed. On the bottom of the screen below the plots and important terms is the model equation in terms of coded values.
Figure C-21. Output of analysis task 8 (for response RCT)

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 C-22.

Figure C-22. Output of analysis task 9. Picture. This figure shows an example of the output from analysis task 9, numerical optimization. The output consists of four lines of text which explain the units for each component, followed by a table listing the variables (factors) in the first column and the corresponding optimum setting from numerical optimization in the second column.
Figure C-22. Output of analysis task 9

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. Picture. This figure is a calculator for predicting responses for selected input variables. The figure consists of a matrix of 10 rows and up to 11 columns. Each row represents a particular combination of input variables. The first five columns of a row are for user entry of variables (W/C, silica fume, etcetera) in appropriate units (W/C by mass, others by volume fraction). If fewer than five variables are used, there will be fewer columns for user entry. The following columns (up to six) contain the predicted total score and responses (up to five, including cost) for the particular set of inputs.
Figure C-23. Calculator for predicting responses based on models

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. Picture. This figure presents a summary screen for the COST analysis. First, the variables examined are listed by name, followed by the responses. This information is followed by notes on the units used for particular variables. Underneath the notes is a table containing the best settings for each of three optimization situations: mean value, individual run, and numerical optimization. The rows of the table are the variables and the columns are the optimization options. The table contains the variable settings corresponding to each optimization scenario.
Figure C-24. Example summary returned by the COST system

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.

 

Previous | Table of Contents


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration