annals of the university of petroşani ∼ economics ∼ vol. xi - part i ...

156 Muntean, C.; Hauer, I.; Butuza, A.
The hierarchy of variants is given by the descending values of the aggregation
function (Aggr).
2.5. The TOPSIS method
The TOPSIS method (Technique for Order Preference by Similarity to Ideal
Solution) is based on the idea that the optimum variant must have the minimum
distance to the ideal solution. The steps of the TOPSIS method are:
- Step 1. We build the normalized matrix R=(r ij ), i=1,...,m, j=1,...,n;
- Step 2. We build the weighted normalized matrix V=(v ij ), i=1,...,m, j=1,...,n, where
v ij =
p
n
∑
j
j = 1
r
ij
p
j
(8)
Step 3. We calculate the ideal solution A and the ideal negative solution B,
defined as:
A= (a 1 , a 2 , ..., a n ), B= (b 1 , b 2 , ..., b n )
where:
⎪⎧
max vij
, if the criterion C j is max
a j =
1≤i≤m
⎨
(9)
min vij
, if the criterion C j is min
⎪⎩ 1≤i≤m
⎪⎧
max vij
, if the criterionC j is min
b j =
1≤i≤m
⎨
(10)
min vij
, if the criterionC j is max
⎪⎩ 1≤i≤m
Step 4. We calculate the distance between the solutions:
n
S i =
∑ ( v ij
− aj )
j = 1
2 , i = 1, 2, … ,m; (11)
n
T i = 2
∑ ( v
ij
− b
j
) , i = 1, 2, ... , m; (12)
j = 1
Step 5. We calculate the relative nearness from the ideal solution:
Ci =
S
i
T
i
+ T
i
(13)
Step 6. We make a classification on the assemblage V according to the
descending values of C i obtained in step 5.