4 The Technique For Order of Preference by Similarity …
29
A
−
= (v
−
1 , v
−
2 , . . . ., v
−
n ) =
min
i
v i j | j ∈ I
,
max
i
v i j | j ∈ J
(4.6)
where I defined with the benefit criteria and J defined with the cost criteria, i =
1, . . . , m; j = 1, . . . , n.
Step 5 Obtaining the separation values of the alternatives and PIS and separation
values of the alternatives and NIS.
In this step several distance metrics can be used. Every alternative from the PIS
is separated based on the following formula;
d
+
i =
⎛
⎝
n
j=1
(v i j − v
+
j )
p
⎞
⎠
1/ p
, i = 1, 2, . . . , m.
(4.7)
Every alternative from the NIS is separated based on the following formula;
d
−
i =
⎛
⎝
n
j=1
(v i j − v
−
j )
p
⎞
⎠
1/ p
, i = 1, 2, . . . , m.
(4.8)
where p ≥ 1. The most commonly used distance metric in this step is the traditional
n-dimensional Euclidean metric (where p = 2) as given in Formulas 4.9 and 4.10.
d
+
i =
n
j=1
v i j − v
+
j ) 2 , i = 1, 2, . . . , m,
(4.9)
d
−
i =
n
j=1
v i j − v
−
j ) 2 , i = 1, 2, . . . , m.
(4.10)
Step 6 Calculating the relative closeness to the PIS.
The relative closeness of the i-th alternative A J concerning A
+ is defined as;
R i =
d
−
i
d
−
i + d
+
i
,
(4.11)
where 0 ≤ R i ≤ 1, i = 1, 2, . . . , m.
In this technique, the alternate closer to the PIS and the further to the NIS should
be preferred (Assari et al. 2012). Corresponding to this argument, the alternatives
are ranked based on the closeness to the PIS. The alternative with a higher R i is the
alternative closest to the PIS. The ranking results can be obtained accordingly.