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By Meilin Wen
This publication is meant to provide the milestones within the development of doubtful information envelopment research (DEA). bankruptcy 1 offers a few easy advent to doubtful theories, together with likelihood idea, credibility concept, uncertainty conception and likelihood concept. bankruptcy 2 provides a complete evaluate and dialogue of uncomplicated DEA versions. The stochastic DEA is brought in bankruptcy three, within which the inputs and outputs are assumed to be random variables. to procure the likelihood distribution of a random variable, loads of samples are had to observe the information inference procedure. bankruptcy four and five offer doubtful DEA ways to review the DMUs with constrained or inadequate statistical info, named fuzzy DEA and unsure DEA. in an effort to evaluation the DMUs within which uncertainty and randomness look concurrently, the hybrid DEA in line with likelihood idea is gifted in bankruptcy 6.
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Extra info for Uncertain Data Envelopment Analysis
Sample text
X/ follows from the definition of chance distribution immediately. 31 (Liu [15]). The uncertain random variables to be independent if ( Ch n \ ) . i " 2 Bi / D E iD1 for any Borel sets B1 ; B2 ; n ^ 1; 2; ; n are said # Mf i . 90) iD1 ; Bn of real numbers. 35. Let 1 ; 2 ; ; n be independent uncertain random variables and f1 ; f2 , ; fn measurable functions. Then f1 . 1 /; f2 . 2 /; ; fn . n / are independent uncertain random variables. Proof. fi . i / 2 Bi / D Ch iD1 " DE ) n \ i 2f iD1 n ^ M ˚ i.
Proof. For any sets B1 ; B2 ; ; Bm of <, we have ( m ) ( m \ \ Cr ffi . Bi /g 1 i iD1 D min Crf 1ÄiÄm Thus f1 . 1 /; f2 . 2 /; ; fm . Bi /g D min Crffi . i / 2 Bi g: 1ÄiÄm m/ are independent fuzzy variables. 21 (Extension Principle of Zadeh). Let 1 ; 2 ; ; n be independent fuzzy variables with membership functions 1 ; 2 ; ; n , respectively, and f W 23 (Peng Iwamura Theorem [21]: Sufficient and Necessary Condition). A function ˆ W < ! x/ Á 1. Proof. It is obvious that an uncertainty distribution ˆ is a monotone increasing function. x/ 6Á 1 follow from the asymptotic theorem immediately. x/ 6Á 1. We will prove that there is an uncertain variable whose uncertainty distribution is just ˆ. Let C be a collection of all intervals of the form . b; 1/, ; and <. We define a set function on < as follows: Mf. b/; Mf