Table 1 Decision results based on the NLNWAA operator by choosing different indeterminate ranges for I in NLNs

I ∈ [−0.7, 0]

E(\( \bar{l}_{1} \)) = 0.6399, E(\( \bar{l}_{2} \)) = 0.6532, E(\( \bar{l}_{3} \)) = 0.6896, E(\( \bar{l}_{4} \)) = 0.6955

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.5, 0]

E(\( \bar{l}_{1} \)) = 0.6479, E(\( \bar{l}_{2} \)) = 0.6576, E(\( \bar{l}_{3} \)) = 0.6978, E(\( \bar{l}_{4} \)) = 0.7038

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.3, 0]

E(\( \bar{l}_{1} \)) = 0.6559, E(\( \bar{l}_{2} \)) = 0.6620, E(\( \bar{l}_{3} \)) = 0.7060, E(\( \bar{l}_{4} \)) = 0.7122

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [−0.1, 0]

E(\( \bar{l}_{1} \)) = 0.6640, E(\( \bar{l}_{2} \)) = 0.6665, E(\( \bar{l}_{3} \)) = 0.7142, E(\( \bar{l}_{4} \)) = 0.7205

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I = 0

E(\( \bar{l}_{1} \)) = 0.6680, E(\( \bar{l}_{2} \)) = 0.6687, E(\( \bar{l}_{3} \)) = 0.7183, E(\( \bar{l}_{4} \)) = 0.7247

u ₄ ≻ u ₃ ≻ u ₂ ≻ u ₁

I ∈ [0, 0.1]

E(\( \bar{l}_{1} \)) = 0.6720, E(\( \bar{l}_{2} \)) = 0.6709, E(\( \bar{l}_{3} \)) = 0.7224, E(\( \bar{l}_{4} \)) = 0.7288

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂

I ∈ [0, 0.3]

E(\( \bar{l}_{1} \)) = 0.6801, E(\( \bar{l}_{2} \)) = 0.6753, E(\( \bar{l}_{3} \)) = 0.7306, E(\( \bar{l}_{4} \)) = 0.7372

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂

I ∈ [0, 0.5]

E(\( \bar{l}_{1} \)) = 0.6881, E(\( \bar{l}_{2} \)) = 0.6797, E(\( \bar{l}_{3} \)) = 0.7388, E(\( \bar{l}_{4} \)) = 0.7455

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂

I ∈ [0, 0.7]

E(\( \bar{l}_{1} \)) = 0.6961, E(\( \bar{l}_{2} \)) = 0.6842, E(\( \bar{l}_{3} \)) = 0.7470, E(\( \bar{l}_{4} \)) = 0.7538

u ₄ ≻ u ₃ ≻ u ₁ ≻ u ₂