Table 2 Decision results based on the NLNWGA operator by choosing different indeterminate ranges for I in NLNs

I ∈ [−0.7, 0]

E(\( \bar{l}_{1} \)) = 0.6285, E(\( \bar{l}_{2} \)) = 0.6470, E(\( \bar{l}_{3} \)) = 0.6887, E(\( \bar{l}_{4} \)) = 0.6956

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.5, 0]

E(\( \bar{l}_{1} \)) = 0.6369, E(\( \bar{l}_{2} \)) = 0.6518, E(\( \bar{l}_{3} \)) = 0.6968, E(\( \bar{l}_{4} \)) = 0.7039

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.3, 0]

E(\( \bar{l}_{1} \)) = 0.6452, E(\( \bar{l}_{2} \)) = 0.6566, E(\( \bar{l}_{3} \)) = 0.7050, E(\( \bar{l}_{4} \)) = 0.7122

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.1, 0]

E(\( \bar{l}_{1} \)) = 0.6535, E(\( \bar{l}_{2} \)) = 0.6614, E(\( \bar{l}_{3} \)) = 0.7131, E(\( \bar{l}_{4} \)) = 0.7205

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I = 0

E(\( \bar{l}_{1} \)) = 0.6577, E(\( \bar{l}_{2} \)) = 0.6638, E(\( \bar{l}_{3} \)) = 0.7172, E(\( \bar{l}_{4} \)) = 0.7247

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [0, 0.1]

E(\( \bar{l}_{1} \)) = 0.6619, E(\( \bar{l}_{2} \)) = 0.6662, E(\( \bar{l}_{3} \)) = 0.7213, E(\( \bar{l}_{4} \)) = 0.7288

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [0, 0.3]

E(\( \bar{l}_{1} \)) = 0.6702, E(\( \bar{l}_{2} \)) = 0.6710, E(\( \bar{l}_{3} \)) = 0.7294, E(\( \bar{l}_{4} \)) = 0.7371

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [0, 0.5]

E(\( \bar{l}_{1} \)) = 0.6786, E(\( \bar{l}_{2} \)) = 0.6758, E(\( \bar{l}_{3} \)) = 0.7376, E(\( \bar{l}_{4} \)) = 0.7454

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂

I ∈ [0, 0.7]

E(\( \bar{l}_{1} \)) = 0.6869, E(\( \bar{l}_{2} \)) = 0.6806, E(\( \bar{l}_{3} \)) = 0.7457, E(\( \bar{l}_{4} \)) = 0.7537

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂