Table 6.

Normalisation techniques

Notation	Techniques	For benefit criteria	For cost criteria	Equation no.	References
N1	Sum-Based Linear Normalisation	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}}{{\sum }_{i=1}^m{X}_{\mathit{ij}}}$	$n_{\mathit{ij}}=\frac{1/x_{\mathit{ij}}}{{\sum }_{i=1}^{m}1/x_{\mathit{ij}}}$	(1)	Wan et al. (2021)
N2	Vector Normalisation	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}}{\sqrt{{\sum }_{i=1}^m{\left|X_{\mathit{ij}}\right|}^2}}$	$n_{\mathit{ij}}=1-\frac{X_{\mathit{ij}}}{\sqrt{{\sum }_{i=1}^m{\left|X_{\mathit{ij}}\right|}^2}}$	(2)	Sayan et al. (2020)
N3	Logarithmic Normalisation	$n_{\mathit{ij}}=\frac{ln{X}_{\mathit{ij}}}{ln\left({\prod }_{i=1}^m{X}_{\mathit{ij}}\right)}$	$n_{\mathit{ij}}=\frac{1-\frac{ln{X}_{\mathit{ij}}}{ln\left({\prod }_{i=1}^m{X}_{\mathit{ij}}\right)}}{m-1}$	(3)	Aytekin (2021; Vafaei et al. (2018)
N4	Maximum–Linear Normalisation	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}$	$n_{\mathit{ij}}=1-\frac{X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}$	(4)	Mishra et al. (2019)
N5	Minimum–Linear Normalisation	$n_{\mathit{ij}}=1-\frac{\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}{X_{\mathit{ij}}}$	$n_{\mathit{ij}}=\frac{\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}{X_{\mathit{ij}}}$	(5)	Aytekin (2021), Vafaei et al. (2018)
N6	0–1 Interval Normalisation Using Max–Min	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}$	$n_{\mathit{ij}}=\frac{\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}{X_{\mathit{ij}}}$	(6)	Wu and Xu (2020)
N7	Jüttler–Körth Normalisation	$n_{\mathit{ij}}=1-\left|\frac{{\text{max}}_i{X}_{\mathit{ij}}-X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}\right|$	$n_{\mathit{ij}}=1-\left|\frac{\underset{i}{\mathit{min}}{X}_{\mathit{ij}}-X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}\right|$	(7)	Aytekin (2021), Vafaei et al. (2018)
N8	Stopp Normalisation	$n_{\mathit{ij}}=\frac{100X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}$	$n_{\mathit{ij}}=\frac{100\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}{X_{\mathit{ij}}}$	(8)	Aytekin (2021), Vafaei et al. (2018)
N9	Nonlinear (Peldschus) Normalisation	$n_{\mathit{ij}}={\left(\frac{X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}\right)}^2$	$n_{\mathit{ij}}={\left(\frac{X_{\mathit{ij}}}{{max}_i{X}_{\mathit{ij}}}\right)}^3$	(9)	Lam et al. (2021)
N10	Weitendorf’s Linear Normalisation	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}-\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}{{\text{max}}_i{x}_{\mathit{ij}}-\underset{i}{\mathit{min}}{x}_{\mathit{ij}}}$	$n_{\mathit{ij}}=\frac{{\text{max}}_i{X}_{\mathit{ij}}-X_{\mathit{ij}}}{{\text{max}}_i{x}_{\mathit{ij}}-\underset{i}{\mathit{min}}{X}_{\mathit{ij}}}$	(10)	Yao (2021)
N11	Z Score Normalisation Depending Optimisation Aspect	$n_{\mathit{ij}}=\frac{X_{\mathit{ij}}-{\mu }_j}{{\sigma }_j}=\frac{X_{\mathit{ij}}-\frac{{\sum }_{i=1}^m{X}_{\mathit{ij}}}{m}}{\sqrt{\frac{{\sum }_{i=1}^m{\left(X_{\mathit{ij}}-{\mu }_j\right)}^2}{m}}}$	$n_{\mathit{ij}}=-\frac{X_{\mathit{ij}}-{\mu }_j}{{\sigma }_j}$	(11)	Aytekin (2021), Vafaei et al. (2018)
N12	Fuzzification Normalisation	Using membership function		(12)	Pamučar et al. (2020), Aytekin (2021), Vafaei et al. (2018)