Matrix Size Vs Random Index (RI) Values

Where aij represents the relative importance of the i-th criterion with respect to the j-th criterion. The pairwise comparison is conducted based on SALTY'S Ratio Scale as presented in Table 1. To illustrate, if two of the criteria for comparison are safety and mobility where the respondent has chosen the value ‘5’ in favor of safety (see Figure 1) then it suggests that the respondent strongly prefers safety over mobility.

Intensity of importance Definition Explanation

  • 1 Equal importance Two items contribute equally to the objective
  • 3 Moderate importance Experience suggests that one be slightly favored over the other
  • 5 Strong importance Experience suggests that one be strongly favored over the other
  • 7 Very strong importance Item strongly favored and its priority demonstrated in practice
  • 9 Absolute importance Importance of one over another affirmed on highest possible order
  • 2,4,6,8 Intermediate values Used to represent compromise between priorities listed above
  • Mobility 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

The calculation of normalizing the attributes involves each of the values in equation (1) being divided by summation of its columns. After that overall average of the row that will determine the ranking of the alternatives is needed to be taken and the outcome is called the priority vectors. If the criteria are represented as a matrix then this priority vector represents the eigenvector of that matrix. From this calculation it is possible to obtain the eigenvalues which will represent the consistency ratio (CR) of the data. This eigenvalue is important to determine the trustworthiness of the data. The eigenvalue is a simple root of the following equation:

  • A×w= ?_max×w (2)

where, A is a matrix of Equation 1, w is the eigenvalue or the local priority vector, ?_max is the largest eigenvalue of matrix A.

Step 3: In this step, the consistency ratio (CR), aka eigenvalue is determined as:

CR=(Consistency Index (CI))/(Random Index (RI)) (3)

Here, CI=( ?_max-n)/(n-1); where, ?_max=maximum eigen value;n=size of the matrix and random index (RI) is such values that have been empirically studied by Saaty(1980) and it has been scaled as shown in Table 2:

  • Matrix, size, n 1 2 3 4 5 6 7 8 9 10
  • Random index, RI 0 0 0.52 0.89 1.11 1.25 1.34 1.41 1.45 1.49
  • (Here, matrix size, n values represent 1x1, 2x2 , … , n x n matrix)

Here, CR assists in determining how consistent the judgements are. A CR0.1 then the responses are dubious triggering the need to revise the data set, revise the pair wise comparison information, and reconstruct the matrix. An acceptable CR value assists to ensure decision-maker reliability in determining the priorities of a set of components (Kumar et al., 2009). The higher CR value acceptance (CR value >0.1) is contingent upon the type of the study, the out coming priorities and the required accuracy (Goepel, 2017).

Step 4: This step consists of deriving relative priorities of the alternatives with respect to each criterion. These priorities are valid only with respect to each speci?c criterion which is known as local priorities (w). For this purpose, a pairwise comparison (using the numeric scale from Table 1) of all the alternatives is conducted with respect to each criterion. The local priority values (w) are next multiplied with weightage factor (µ) to convert them into global priority factors, wglobal. as follows:

  • w_global=µ×w (4)

Here, µ is a matrix of the alternative with respect to each criteria using Saaty's 1 to 9 scale which is obtained by following the same procedure stated above, and w is local vectors that was determined from Step 2 when the overall average of the row was taken after normalizing A.

This study employed various packages of open source software “R” for text mining and topic modelling. The “topicmodels” package (Grun and Hornik, 2011) is used for LDA, “tm” for text mining (Feinerer & Hornik, 2015), and “wordcloud” to visualize the word clouds (Fellows, 2014).

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Matrix size vs random index (RI) values. (2021, Dec 29). Retrieved March 2, 2024 , from

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