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Author: Admin | 2025-04-28
L 3 ( p ) , α 1 ( k ) , α 2 ( k ) and α 3 ( k ) are the probabilities of the k t h linguistic terms in L 1 ( p ) , L 2 ( p ) and L 3 ( p ) respectively. Then, the generalized hybrid distance has the following properties:(1) D g h ( L 1 ( p ) , L 2 ( p ) ) ≥ 0 ;(2) D g h ( L 1 ( p ) , L 2 ( p ) ) = 0 ⇔ L 1 ( p ) = L 2 ( p ) ;(3) If L 1 ( p ) L 2 ( p ) L 3 ( p ) , then D g h ( L 1 ( p ) , L 2 ( p ) ) D g h ( L 1 ( p ) , L 3 ( p ) ) , D g h ( L 2 ( p ) , L 3 ( p ) ) D g h ( L 1 ( p ) , L 3 ( p ) ) .The proof of Theorem 1 is given in Appendix A. 2.3. Probabilistic Linguistic CPT 2.3.1. Classical CPTTo better retain the true evaluation information of DMs, probabilistic fusion is performed using CPT. CPT [35] is an improved version of prospect theory (PT) [36] to address stochastic dominance proposed by Tversky et al. in 1992, which well explains phenomena such as stochastic dominance, and its measure of the total value of a prospect through a value function and probability weights. The forms are shown as follows:Combined prospect value: V ( x ) = ∑ i = 0 n v ( x i ) w ( p i ) (8) CPT asserts that there exist a strictly increasing weighted value function v ( x i ) . The value function v ( x i ) is defined on the deviations from a reference point, which represents the behavior of the DMs and can be expressed as follows.Value function: v ( x i ) = ( x i − b ) ξ , x i ≥ b − θ ( b − x i ) β , x i b (9) The key difference between CPT and PT is that the weight function used in CPT is no longer a
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