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B. Uzun et al.
Step 1 Construct a decision matrix and define the importance weights of the
criteria.
The decision matrix X = X i j and a weighting vector W = [w 1 , w 2 , . . . , w n ] are
chosen. Where X i j ∈ R, W j ∈ R and w 1 + + . . . + w n = 1.
Function criteria can either be a “benefit function” (more benefit better alternative)
or a “cost function” (low cost better alternative).
Step 2 Obtaining the normalized decision matrix.
In this step, the different kinds of dimensions of the criteria are converted into
one dimension that allows comparison between all criteria. To convert all data to
normalized form, each element of the matrix X must be converted, since most criteria
are usually measured in different units. The normalization of these values can be
done using one of the normalization formulas. The normalized values of the decision
matrix (n i j ) can be obtained with one of the following formulas:
n i j =
X i j
m
i=1 X
2
i j
(4.1)
n i j =
X i j
max
i
X i j
(4.2)
n i j =
⎧
⎪ ⎨
⎪ ⎩
x i j −min
i
x i j
max
i
x i j −min
i
x i j
max
i
x i j −x i j
max
i
x i j −min
i
x i j
(4.3)
Step 3 Obtaining the weighted normalized decision matrix.
Method to obtain the weighted normalized value of the decision matrix
v i j
;
v i j = w j n i j
(4.4)
where i = 1, . . . , m; j = 1, . . . , n. And w j is the weight of the j-th criteria where
n
j=1 w j = 1.
Step 4 Identify the positive ideal and negative ideal solutions.
Here, the PIS on each criterion (the best performance) and the NIS (the worst
performance) are identified. The PIS maximizes the benefit criteria and minimizes
the cost criteria, while the NIS maximizes the cost criteria and minimizes the benefit
criteria.
The PIS A
+ can be obtained with the following formula:
A
+
= (v
+
1 , v
+
2 , . . . , v
+
n ) =
max
i
v i j | j ∈ I
,
min
i
v i j | j ∈ J
(4.5)
And oppositely the NIS A
− can be obtained with the following formula: