By Zeshui Xu
ISBN-10: 3319047108
ISBN-13: 9783319047102
ISBN-10: 3319047116
ISBN-13: 9783319047119
This booklet offers the readers with a radical and systematic advent to hesitant fuzzy thought. It provides the newest examine effects and complicated equipment within the box. those comprises: hesitant fuzzy aggregation recommendations, hesitant fuzzy choice family members, hesitant fuzzy measures, hesitant fuzzy clustering algorithms and hesitant fuzzy multi-attribute determination making tools. considering its creation by way of Torra and Narukawa in 2009, hesitant fuzzy units became an increasing number of renowned and feature been used for quite a lot of purposes, from decision-making difficulties to cluster research, from scientific prognosis to body of workers appraisal and knowledge retrieval. This e-book bargains a entire file at the state of the art in hesitant fuzzy units idea and functions, aiming at turning into a reference consultant for either researchers and practitioners within the quarter of fuzzy arithmetic and different utilized learn fields (e.g. operations examine, info technology, administration technological know-how and engineering) characterised via doubtful ("hesitant") details. due to its readability and self contained motives, the booklet is usually followed as a textbook from graduate and complicated undergraduate students.
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Hesitant Fuzzy Sets Theory - download pdf or read online
This ebook offers the readers with an intensive and systematic advent to hesitant fuzzy thought. It offers the newest study effects and complex equipment within the box. those comprises: hesitant fuzzy aggregation suggestions, hesitant fuzzy choice kin, hesitant fuzzy measures, hesitant fuzzy clustering algorithms and hesitant fuzzy multi-attribute choice making equipment.
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Proof. In the following, we prove (1), (3), (5) and (7), others can be proven similarly: (1) ( h1 h2 ) ⊕ h3 = = = = γ 1∈h1 ,γ 2 ∈h2 max{γ 1 , γ 2 } ⊕ h3 s −1 ( s(max{γ 1 , γ 2 }) + s(γ 3 ) ) s −1 ( max {s(γ 1 ) + s(γ 3 ), s(γ 2 ) + s(γ 3 )} ) γ 1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 γ 1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 γ1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 ( max {s −1 ( s(γ 1 ) + s(γ 3 ) ) , s−1 ( s(γ 2 ) + s(γ 3 ) )}) = {s−1 ( s(γ 1 ) + s(γ 3 ) )} {s−1 ( s(γ 2 ) + s(γ 3 ) )} γ1∈h1 ,γ 3∈h3 γ 2∈h2 ,γ 3∈h3 = (h1 ⊕ h3 ) (h2 ⊕ h3 ) .
Let HFEs having the weight vector i = 1, 2, , n , and n w i =1 i hi ( i = 1, 2, , n ) be a collection of w = ( w1 , w2 wn )Τ such that wi ∈ [0,1] , = 1 , λ > 0 . 2 Hesitant Fuzzy Aggregation Operators Especially, if 37 λ = 1 , then the GHFWG operator becomes the HFWG operator; Τ 1 1 1 If w = , , , , then the GHFWG operator reduces to the GHFG n n n operator. 23 (Xia and Xu 2011a). For a collection of HFEs hi ( i = 1, 2, , n ), w = ( w1 , w2 , , wn )Τ that i = 1, 2, , n , and is n w i =1 i the weight vector such wi ∈ [0,1] , = 1 , λ > 0 , then GHFWG λ (h1 , h2 , , hn ) ≤ HFWA ( h1 , h2 , , hn ) Proof.
Let (1) (2) (3) (4) (5) (6) (7) (8) h1 , h2 and h3 be three HFEs, then (h1 h2 ) ⊕ h3 = (h1 ⊕ h3 ) (h2 ⊕ h3 ) . (h1 h2 ) ⊕ h3 = (h1 ⊕ h3 ) (h2 ⊕ h3 ) . (h1 h2 ) ⊗ h3 = (h1 ⊗ h3 ) (h2 ⊗ h3 ) . (h1 h2 ) ⊗ h3 = (h1 ⊗ h3 ) (h2 ⊗ h3 ) . h1 ⊕ (h2 h3 ) = (h1 ⊕ h2 ) (h1 ⊕ h3 ) . h1 ⊕ (h2 h3 ) = (h1 ⊕ h2 ) (h1 ⊕ h3 ) . h1 ⊗ (h2 h3 ) = (h1 ⊗ h2 ) (h1 ⊗ h3 ) . h1 ⊗ (h2 h3 ) = (h1 ⊗ h2 ) (h1 ⊗ h3 ) . Proof. In the following, we prove (1), (3), (5) and (7), others can be proven similarly: (1) ( h1 h2 ) ⊕ h3 = = = = γ 1∈h1 ,γ 2 ∈h2 max{γ 1 , γ 2 } ⊕ h3 s −1 ( s(max{γ 1 , γ 2 }) + s(γ 3 ) ) s −1 ( max {s(γ 1 ) + s(γ 3 ), s(γ 2 ) + s(γ 3 )} ) γ 1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 γ 1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 γ1∈h1 ,γ 2 ∈h2 ,γ 3 ∈h3 ( max {s −1 ( s(γ 1 ) + s(γ 3 ) ) , s−1 ( s(γ 2 ) + s(γ 3 ) )}) = {s−1 ( s(γ 1 ) + s(γ 3 ) )} {s−1 ( s(γ 2 ) + s(γ 3 ) )} γ1∈h1 ,γ 3∈h3 γ 2∈h2 ,γ 3∈h3 = (h1 ⊕ h3 ) (h2 ⊕ h3 ) .
Hesitant Fuzzy Sets Theory by Zeshui Xu
by Thomas
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