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27 Table 24. Example of the relative importance of the test methods characteristics. Reproducibility Interpret Data Repeatability Applicability Ease of Use Characteristics of Portability Readiness Accuracy Test Methods Price Repeatability 1 1 0.33 1 1 1 1 1 1 Reproducibility 1 1 0.33 1 1 1 1 1 1 Accuracy 3 3 1 5 5 5 5 5 5 Cost 1 1 0.2 1 1 1 1 1 1 Readiness 1 1 0.2 1 1 1 1 1 1 Interpret Data 1 1 0.2 1 1 1 1 1 1 Ease of Use 1 1 0.2 1 1 1 1 1 1 Portability 1 1 0.2 1 1 1 1 1 1 Applicability 1 1 0.2 1 1 1 1 1 1 Readiness/Portability: The scale for readiness reflects the aggregate types. A few examples are provided to highlight preference for a test method that has been used by research the process for ranking the test methods. and testing laboratories and thus methods that are not avail- able commercially are considered slightly less desirable than Fine Aggregate Angularity those that are available. However, this point is not highly emphasized in the scale (the maximum possible ratio is AHP was used to rank the test methods that measure fine only 5) because any of the methods can be made available aggregate angularity: uncompacted void content of fine aggre- commercially in the future. The same applies for porta- gate (UCVCF), compacted aggregate resistance (CAR), Cam- bility, as the portability of those methods that are given sizer, Buffalo Wire Works (PSSDA-Small), and AIMS. In this a scale of "3 Not portable" can be improved with some example, the same weights (1) were assigned for all the charac- design changes. teristics in the second level (i.e., characteristics were consid- Interpretation of Data and Ease of Use: The values assigned ered equally important). This means that all cells in Table 27 in Table 22 are based on current knowledge of the test will have a value of 1. methods regarding their use in routine analysis of aggregates. A pairwise comparison of all test methods according to Except for the methods labeled (4:difficult), technical train- one characteristic was then conducted using numerical ratings ing can improve the assigned value from (3:intermediate) selected from Table 23. These ratings are used in Table 28 to (2:easy) or even (1:very easy) indicating that the change in order to compare a test method from the horizontal list from 3 to 4 is less desirable than the change from 1 to 3. to that of the vertical list based on the characteristics under Applicability to Measure Different Aggregate Types and consideration. Sizes: Test methods are expected to measure all aggregate Once the values in Tables 27 and 28 are assigned, the next types and sizes. If the method fails to measure some sizes step consists of the computation of priority lists of test methods or some aggregate types, or both, its applicability rating for each of the desirable characteristics. In mathematical terms, should be reduced. The values assigned for the applicability the principal eigen vector is computed for each matrix which of test method to measure different aggregate types and gives the vector of priority ordering. Saaty (21) proposed sizes listed in Table 22 are based on current knowledge and some crude estimates that can be easily followed to calculate experience with the test methods. The assigned values assume these vectors. One good estimate method is to divide the ele- that it is weakly more important (assigned a value of 3) to ments of each column in the matrix by the sum of that column have a method that can measure all aggregate types and sizes (i.e., normalize the column). Then elements in each resulting than a method that can measure all aggregate types but not row are added then divided by the number of elements in the all aggregate sizes or to have a method that can measure some row. This is a process of averaging over the normalized column. aggregate sizes for all aggregate types than a method that can The resulting priority vectors from each matrix in Table 29 measure all sizes for some aggregate types. It is considered are then combined to create a matrix that represents priority moderately more important (assigned a value of 5) to have of test method by each characteristic. In order to obtain the a method that can measure all aggregate types and sizes than overall ranking of the test methods, the priority matrix of a method that can measure all aggregate sizes but not all the methods by each characteristic will be multiplied by the

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28 Table 25. Weights that compare test methods based on each of the characteristics. Characteristic Criterion Comparison Scale 1 3 5 7 9 Repeatability/ 2:1 X Reproducibility 3:2 X 3:1 X Accuracy Coarse-Shape 2:1 X (Ratio of R2 groups) 3:1 X 3:2 X 4:1 X 4:2 X 4:3 X Accuracy Coarse- 2:1 X Irregularity (Ratio of R2 3:1 X groups) 3:2 X 4:1 X 4:2 X 4:3 X Accuracy Coarse-Texture 2:1 X (Ratio of R2 groups) 3:1 X (Rankings) 3:2 X 4:1 X 4:2 X 4:3 X Accuracy Fine-Angularity 2:1 X (Ratio of R2 groups) 3:1 X 3:2 X 4:1 X 4:2 X 4:3 X Price (Ratio of Cost) <6 X >6 <20 X >20 <50 X >50 <80 X >80 X Readiness 2:1 X 3:2 X 3:1 X Portability 2:1 X 3:2 X 3:1 X Data Interpretation 2:1 X 3:2 X 3:1 X 4:3 X 4:2 X Ease of Use 2:1 X 3:2 X 3:1 X 4:3 X 4:2 X Applicability 2:1 X 3:2 X 3:1 X priority vector of the characteristics resulting from Table 27. Table 26. Accuracy categories based In other words, the overall ranking of a method can be obtained on R2 values. by multiplying the weight indicating the rank of a test method with respect to the characteristic by the weight of that char- R2 Category > 0.70 1 acteristic then add them up for all characteristics. The result- 0.6 0.7 2 ing priority vectors and the overall ranking of test methods 0.5 0.6 3 used to measure fine aggregate angularity are presented in < 0.5 4 Table 29.