By Meilin Wen

This e-book is meant to give the milestones within the development of doubtful info envelopment research (DEA). bankruptcy 1 offers a few uncomplicated advent to doubtful theories, together with chance thought, credibility idea, uncertainty concept and probability idea. bankruptcy 2 offers a finished assessment and dialogue of uncomplicated DEA versions. The stochastic DEA is brought in bankruptcy three, during which the inputs and outputs are assumed to be random variables. to procure the chance distribution of a random variable, loads of samples are had to follow the information inference process. bankruptcy four and five supply doubtful DEA easy methods to assessment the DMUs with constrained or inadequate statistical info, named fuzzy DEA and unsure DEA. with a purpose to assessment the DMUs within which uncertainty and randomness look concurrently, the hybrid DEA according to likelihood thought is gifted in bankruptcy 6.

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**Uncertain Data Envelopment Analysis**

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Liu B (2002) Theory and practice of uncertain programming. Physica-Verlag, Heidelberg 6. Liu B (2004) Uncertainty theory. Springer, Berlin 7. Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin 8. Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16 9. Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10 10. Liu B (2009) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin 11. Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty.

19. The fuzzy variables ( Cr m [ 1; 2; ; m are independent if and only if ) f i 2 Bi g D max Cr f iD1 for any sets B1 ; B2 ; 1; 2; ; Bm of <. 2 Credibility Theory 21 1; 2; : : : ; m Proof. 19) is verified. The proof is complete. 20. Let i be independent fuzzy variables and fi W < ! < functions, i D 1; 2; ; m. Then f1 . 1 /; f2 . 2 /; ; fm . m / are independent fuzzy variables. 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 .

A function. Then the membership function of D f . x1 ;x2 ; ;xn / 1ÄiÄn Proof. 14 that the membership function of f . 2Crff . x1 ;x2 ;:::;xn / D 2 D D sup 2 n 91 = D xn g A ^ 1 ; ! D x2 ; : : : ; n D xn g ^ 1 ! 2. 3. a1 ; a2 ; a3 / and a scalar number . a1 ; a2 ; a3 /; if 0 . 4. a1 ; a2 ; a3 ; a4 / and a scalar number is also a trapezoidal fuzzy variable, and ( D . a1 ; a2 ; a3 ; a4 /; if 0 . 18 (Liu and Liu [19]). Let value of is defined by Z EŒ D Z C1 Crf 0 be a fuzzy variable. Then the expected rgdr 0 1 Crf Ä rgdr provided that at least one of the two integrals is finite.