. 2022 Jan 27;55(6):4979–5062. doi: 10.1007/s10462-021-10124-x

Table 16.

Employed aggregation and defuzzification equations per FDOSM version

Ver.	Aggregation formula		Defuzzification formula
1st	$Arithmetic mean A_{m (x)} = \frac{\sum_{i = 1}^{n} x_{i}}{n}$ $A_{m (x) =} \frac{\sum (a_{f} + a_{m} + a_{l}) (b_{f} + b_{m} + b_{l})}{n} \frac{(c_{f} + c_{m} + c_{l})}{n}$	(50)	$\frac{(a + b + c)}{3}$	(51)
2nd	$A M_{(x)} = \frac{\sum_{i = 1}^{n} x_{i}}{n}$ $A M_{(x)} = \frac{\sum (a_{f} + a_{m} + a_{l}) (b_{f} + b_{m} + b_{l}) (c_{f} + c_{m} + c_{l})}{n}$	(52)
	$G M_{(x)} = {(\prod_{i = 1}^{n} x_{i})}^{\frac{1}{n}}$ $G M_{(x)} = {(\prod (a_{f} \times a_{m} \times a_{l}) (b_{f} \times b_{m} \times b_{l}) (c_{f} \times c_{m} \times c_{l}))}^{1 / n}$	(53)
	$H M_{(x)} = {(\frac{\sum_{i = 1}^{n} x_{i}^{- 1}}{n})}^{- 1}$ $H M_{(x)} = {(\frac{\sum {(a_{f} + a_{m} + a_{l})}^{- 1} {(b_{f} + b_{m} + b_{l})}^{- 1} {(c_{f} + c_{m} + c_{l})}^{- 1}}{n})}^{- 1}$	(54)
	$R M_{(x)} = \sqrt{\frac{1}{n} \times \sum_{i = 1}^{n} {(x_{i})}^{2}}$ $R M_{(x)} = \sqrt{\frac{1}{n} \times \sum_{i = 1}^{n} {(a_{f} + a_{m} + a_{l})}^{2} {(b_{f} + b_{m} + b_{l})}^{2} {(c_{f} + c_{m} + c_{l})}^{2}}$	(55)
3rd	$\begin{matrix} \approx \approx_{A 1} = (A_{1}^{U}, A_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{λ 3}^{U}, a_{14}^{U} ; H_{1} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{1}^{U}))) \\ ((a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L} ; H_{1} (A_{1}^{L}), H_{2} (A_{1}^{L}))) and \\ \approx \approx_{A 2} = (A_{2}^{U}, A_{2}^{L}) = ((a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U} ; H_{1} (A_{2}^{U}), H_{2} (A_{2}^{U}))) \\ ((a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L} ; H_{1} (A_{2}^{L}), H_{2} (A_{2}^{L}))) the addtion as follow \\ \underset{A_{1}}{\approx} \oplus \underset{A_{2}}{\approx} \\ = ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) \oplus ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) = ((a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U})) \\ (+ a_{24}^{U} ; m i n (H_{1} ({\tilde{A}}_{1}^{U}), H_{1} ({\tilde{A}}_{2}^{U})), m i n (H_{2} (\tilde{A_{1}^{U}}), H_{2} (\tilde{A_{2}^{U}}))) \\ (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L} ; m i n (H_{1} ({\tilde{A}}_{1}^{L}), H_{1} ({\tilde{A}}_{2}^{L}))) \\ ((m, i, n, (H_{2} (A_{1}^{L}), H_{2} (A_{2}^{L})))) \end{matrix}$	(56)	$(\tilde{V (A_{i}^{U})} = \frac{1}{n (n - 1)} (\sum_{k = 1}^{n} p \tilde{(A_{i}^{U})} \geq \tilde{A_{k}^{U}}) + \frac{n}{2} - 1)$ $(\tilde{V (A_{i}^{L})} = \frac{1}{n (n - 1)} (\sum_{k = 1}^{n} p \tilde{(A_{i}^{L})} \geq \tilde{A_{k}^{L}}) + \frac{n}{2} - 1)$ $\underset{A_{i}}{=} \frac{V (A_{i}^{U}) + \tilde{V (A_{i}^{L})}}{2}$	(57)
4th	$PFWA (p (m_{1}), p (m_{2}), \dots, p (m_{n})) = (\sum_{i = 1}^{n} w_{i} μ_{p} (m_{i}), \sum_{i = 1}^{n} w_{i} v_{p} (m_{i}))$	(58)	$α = (μ_{z}, v_{α})$ $, s (α) = μ_{α}^{2} - v_{α}^{2} h (α) = μ_{α}^{2} + v_{z}^{2}$	(59)
5th	$T - SWAM ({\tilde{p}}_{1}, {\tilde{p}}_{2}, \dots, {\tilde{p}}_{n}) = \{{[1 - \prod_{i = 1}^{n} (1 - μ_{ρ_{i}}^{2}) w_{i}]}^{1 / T}, \prod_{i = 1}^{n} v_{{\bar{p}}_{i}}^{w_{i}} \cdot {[\prod_{i = 1}^{n} (1 - μ_{{\tilde{p}}_{i}}^{2}) w_{i} - \prod_{i = 1}^{n} {(1 - μ_{{\tilde{β}}_{i}}^{2} - s_{β_{i}}^{2})}^{w_{i}}]}^{1 / T}\}$	(60)	$Score (\tilde{p}) = μ_{\tilde{p}}^{T} - s_{\tilde{p}}^{T}$	(61)
6th	$q - RoFWA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = ⟨ {(1 - \prod_{k = 1}^{n} {(1 - μ_{k}^{q})}^{w_{k}})}^{\frac{1}{q}}, \prod_{k = 1}^{n} v_{k}^{w_{k}} ⟩$	(62)	$S_{k} = μ_{k}^{q} - v_{k}^{q}, where q \geq 1$	(63)