2.2.2. AHP Technique
The AHP technique was proposed by Saaty [
24]. The pairwise matrix
A is used to compare the set of
n alternatives according to the relative importance of the weights.
In Equation (2),
a1,
a2, …,
an represent the criteria. The relative significance of the two criteria are ranked using the digits from 1 to 9 [
25], where 1 stands for equally significant, 3 for somewhat more significant, 5 for intensely significant, 7 for demonstrably more significant, 9 for absolutely more significant and 2, 4, 6, 8 stands for a compromise between slightly different judgments. The comparative weights are obtained by determining the eigenvector
w concerning
that satisfies
Aw =
. Here,
is the highest eigenvalue of matrix
A. The consistency index (CI) and the consistency ratio (CR) are calculated from Equations (3) and (4) to ascertain the accuracy of the comparative weights and consistency of the subjective perception. Here,
n is a criteria number, the value of CI should be less than 0.1 for the results to be confident, and the random consistency index (RI) should be below 0.1 for the valid results.
2.2.4. Exprom2 Technique
The updated version of the Promethee II technique is known as the Exprom2 technique, which is derived from ideal and anti-ideal solutions given in Equations (10) and (11) [
26]:
In this xij indicates the performance measure of the ith alternative concerning criterion j and rij is the normalized value of the xij.
The pairwise calculation is done to get the difference in criteria value (dj).
Equation (12) is used to calculate the preference function to measure the extent to which the alternative
i dominate over the alternative
i/ for the
jth criteria.
Here
is the value of
jth criteria and
ith alternative. To get the weights by the entropy technique, the normalized decision matrix
Pij is estimated by Equation (13) given by Gowda and Jayaramaiah [
27].
Equation (14) is used to calculate the weak preference index
.
In Equation (14), the wj is the weight of the jth criterion derived from the compromised weighting method.
The definition of the strict preference function is given by Equations (15) and (16).
In this,
dmj is the difference between the ideal and anti-ideal value of the
jth criterion and the
Lj is the limit of the preference.
The total preference index is given by Equation (17).
The positive flow for the
ith alternative is calculated by Equation (18).
The negative flow for the
ith alternative is calculated by the by Equation (19).
Here m is the number of the alternatives.
The total outranking flow for each alternative is calculated by Equation (20).
The best alternative is chosen based on the highest value of .
The technique for order of preference by similarity to ideal solution method (TOPSIS)
The steps followed in the TOPSIS method are:
1. The decision matrix with dimension, mxn is formed by n criteria and m alternatives with the interaction of each criterion and alternative given by .
2. The weight of each criterion is computed by comparing the relative importance of one attribute with the other (Saaty 1980) [
24].
3. The matrix
is now normalized (Equation (21)):
Using the normalization method (Equation (22)):
4. Calculate the weighted normalized decision matrix from Equations (23) and (24) in which,
T is the resultant of matrix operation,
tij is the matrix element corresponding to the
ith row and
jth column:
where
,
is the original weight given to the indicator
5. Determine the worst alternative
(Equation (25)) and the best alternative
(Equation (26)):
In which, has a positive impact, and has a negative impact.
Equation (27) is used to calculate the distance between the target alternative
and the worst condition
.
The alternative
and the best condition
(Equation (28)):
where,
and
and
are the distances from the target option
to the best and worst conditions, respectively.
6. Euclidean distance
is calculated by Equation (29) to determine the similarity to the worst condition:
for the best condition, and; for the worst condition.
7. Ranking of by prioritization.
2.2.5. VIKOR Technique
The VIKOR method gives a compromised solution. The main steps of the VIKOR technique [
28] are:
The decision matrix is used to obtain the best i.e., (xij)max and the worst i.e., (xij)min values of all the criteria.,
The following equations are used to determine the standard parameters of the VIKOR method:
Ei,
Fi, and
Pi respectively.
The values of
Pi are calculated by Equation (32).
Here is the weight of the policy of the majority of the criteria, the range of this can be any value between 0 and 1 and the most common value is 0.5. The maximum and minimum values of Ei and Fi are designated by Ei-max, Ei-min, Fi-max, and F-min respectively.
The criteria weighting and the MADM techniques have been explicitly given in the preceding section and
Figure 2 presents a summary in a flow chart. The abbreviations, mathematical operators, and symbols have been given at the end of the article for quick reference.
Sensitivity analysis deals with the effect of criteria weights obtained by the combined approach (AHP-entropy) on the alternatives. The basic methodology of sensitivity analysis is that the weight of each criterion is interchanged with another. To validate the criteria weighting and compare the results obtained from the MADM techniques, sensitivity analysis was done. A similar methodology has been adopted by Ahmed et al. [
17,
18]. The criteria weights were obtained from a survey among the stakeholders and the recent data obtained from the government sources [
21,
22].