A comparative evaluation of data normalization techniques using different metrics: practical application to a MCDM method

Open Access

Table 1

Normalization techniques.

Codes	Norm. techniques	B	C	References
N1	Sum-Based Linear	$n_{i j} = \frac{x_{i j}}{\sum_{i = 1}^{m} x_{i j}}$	$n_{i j} = \frac{1 / x_{i j}}{\sum_{i = 1}^{m} (1 / x_{i j})}$	[27]
N2	Vector	$n_{i j} = \frac{x_{i j}}{\sqrt{\sum_{i = 1}^{m} {(x_{i j})}^{2}}}$	$\begin{array}{l} n_{i j} \\ = 1 - \frac{x_{i j}}{\sqrt{\sum_{i = 1}^{m} {(x_{i j})}^{2}}} \end{array}$	[28]
N3	Maximum - Linear	$n_{i j} = \frac{x_{i j}}{\max_{i} x_{i j}}$	$n_{i j} = \frac{\min_{i} x_{i j}}{x_{i j}}$	[29]
N4	Linear max min	$n_{i j} = \frac{r_{i j} - r_{j}^{m i n}}{r_{i}^{m a x} - r_{i}^{m i n}}$	$n_{i j} = \frac{r_{i}^{m a x} - r_{i j}}{r_{i}^{m a x} - r_{i}^{m i n}}$	[30]
N5	Peldschus	$n_{i j} = {(\frac{x_{i j}}{\max_{i} x_{i j}})}^{2}$	$n_{i j} = {(\frac{\min_{i} x_{i j}}{x_{i j}})}^{3}$	[28]
N6	Logarithmic	$n_{i j} = \frac{I n x_{i j}}{\ln (\prod_{i = 1}^{m} x_{i j})}$	$\begin{array}{l} n_{i j} \\ = \frac{1 - \frac{I n x_{i j}}{\ln (\prod_{i = 1}^{m} x_{i j})}}{m - 1} \end{array}$	[28]
N7	Stopp	$n_{i j} = \frac{100 x_{i j}}{\max_{i} x_{i j}}$	$n_{i j} = \frac{100 \min_{i} x_{i j}}{x_{i j}}$	[31]
N8	Enhanced accuracy	$\begin{array}{l} n_{i j} \\ = 1 \\ - \frac{r_{j}^{m a x} - r_{i j}}{\sum_{i = 1}^{m} (r_{j}^{m a x} - r_{i j})} \end{array}$	$\begin{array}{l} n_{i j} \\ = 1 \\ - \frac{r_{i j} - r_{j}^{m i n}}{\sum_{i = 1}^{m} (r_{i j} - r_{j}^{m i n})} \end{array}$	[32]
N9	Double	$\begin{array}{l} r_{i j} = \frac{x_{i j}}{x_{j m a x}}, a n d {r^{'}}_{i j} \\ = \frac{x_{j m i n}}{x_{i j}} \end{array}$	$\begin{array}{l} r_{i j} = \frac{x_{j m i n}}{x_{i j}}, a n d {r^{'}}_{i j} \\ = \frac{x_{i j}}{x_{j m a x}} \end{array}$	[33]

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