Normalization Techniques

All the MCDM techniques aim to evaluate and rank a group of alternatives that satisfies a set of given criteria [1]. In MCDM problems, criteria represent a set of independent evaluation attributes that must be satisfied by the considered alternatives. In these problems, each criterion may have different units or dimensions, e.g., kilograms or meters; but all of them have to be normalized/pre-processed to define a common dimensionless range/scale in order to allow their combination in the form of a single score for each alternative. Furthermore, if the normalization technique is not suitable for the given decision problem or for the selected MCDM technique, the best decision solution may often be overlooked [1]. Hence, normalization/pre-processing of data is considered to be an integral part of any decision-making process mainly for the reason that it converts the input information into comparable arithmetical data, thus permitting MCDM techniques to assess and rank the alternatives [2–5]. It has already been pointed out that although the normalization process scales the criteria values to be roughly around equivalent magnitude, different normalization methods can produce different solutions, which may lead in deviation from the initially recommended solutions. Different normalization procedures as recommended for dealing with MCDM problems are enlisted as below:

Linear: Max: 31_1 https://www.w3.org/1998/Math/MathML"> n i j = x i j x i j max   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_2 https://www.w3.org/1998/Math/MathML"> n i j = 1 − x i j x i j max   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where; x_ij is the performance score of i ^th alternatives with respect to j ^th criterion; https://www.w3.org/1998/Math/MathML"> x j i max https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/inline31_351_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the maximum value of the considered criterion; and n_ij is the normalized value of x_ij.

Linear: Max-Min: 31_3 https://www.w3.org/1998/Math/MathML"> n i j = x i j − x i j min x i j max − x i j min   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_4 https://www.w3.org/1998/Math/MathML"> n j i = x i j max − x i j x i j max − x i j min   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where; https://www.w3.org/1998/Math/MathML"> x i j min https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/inline31_352_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> min is the minimum value of the criterion under consideration.

Linear: Sum: 31_5 https://www.w3.org/1998/Math/MathML"> n i j = x i j ∑ j = 1 n x i j   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_6 https://www.w3.org/1998/Math/MathML"> n i j = 1 / x i j ∑ j = 1 n ( 1 / x i j )   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where; n is the number of criteria.

Vector Normalization: 31_7 https://www.w3.org/1998/Math/MathML"> n i j = x i j ∑ j = 1 n x i j 2   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_7_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_8 https://www.w3.org/1998/Math/MathML"> n i j = 1 − x i j ∑ j = 1 n x i j 2   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

Logarithmic Normalization: 31_9 https://www.w3.org/1998/Math/MathML"> n i j = ln ( x i j ) ln ( ∏ j = 1 n x i j )   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_9_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_10 https://www.w3.org/1998/Math/MathML"> n i j = 1 − ln ( x i j ) ln ( ∏ j = 1 n x i j ) n − 1   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_10_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

Non-Linear Normalization: 31_11 https://www.w3.org/1998/Math/MathML"> n i j = ( x i j x i j max ) 2   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_11_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_12 https://www.w3.org/1998/Math/MathML"> n i j = ( x i j min x i j ) 2   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_12_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

Jüttler’s-Körth’s Normalization (JKN): 31_13 https://www.w3.org/1998/Math/MathML"> n i j = 1 − | x i j max − x i j x i j max |   ( for   beneficial   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_13_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 31_14 https://www.w3.org/1998/Math/MathML"> n i j = 1 − | x i j min − x i j x i j min |   ( for   cost   criterion ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math31_14_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>