By Boading Liu

ISBN-10: 3790814903

ISBN-13: 9783790814903

Real-life judgements are typically made within the nation of uncertainty (randomness, fuzziness, roughness, etc.). How can we version optimization difficulties in doubtful environments? How can we resolve those versions? so that it will solution those questions, this ebook offers a self-contained, complete and up to date presentation of doubtful programming idea. It contains quite a few modeling rules, hybrid clever algorithms, and numerous purposes in transportation challenge, stock method, facility place & allocation, capital budgeting, topological optimization, automobile routing challenge, redundancy optimization, and scheduling. Researchers, practitioners and scholars in operations study, administration technology, details technological know-how, procedure technology, and engineering will locate this paintings a stimulating and necessary reference.

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**New PDF release: Theory and Practice of Uncertain Programming**

Real-life judgements tend to be made within the country of uncertainty (randomness, fuzziness, roughness, and so on. ). How can we version optimization difficulties in doubtful environments? How can we resolve those versions? with a purpose to resolution those questions, this ebook presents a self-contained, entire and updated presentation of doubtful programming thought.

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**Extra info for Theory and Practice of Uncertain Programming**

**Example text**

Thus the support E ∗ of the event E is {x1 , x2 , x5 , x6 , x9 , x10 }. 18 The jth constraint gj (x, ξ) ≤ 0 is called an active constraint of the event E if the set of nondegenerate decision variables of gj (x, ξ) and the support E ∗ have nonempty intersection; otherwise it is inactive. 19 Let E be an event hk (x, ξ) ≤ 0, k = 1, 2, · · · , q in the uncertain environment gj (x, ξ) ≤ 0, j = 1, 2, · · · , p. The dependent support of the event E, denoted by E ∗∗ , is defined as the set consisting of all nondegenerate decision variables of hk (x, ξ), k = 1, 2, · · · , q and gj (x, ξ) in the active constraints to the event E.

Set N = 0. Step 2. Generate ω from Ω according to the probability measure Pr. Equivalently, we generate a random vector ξ(ω) according to its probability distribution. Step 3. If gj (x, ξ(ω)) ≤ 0 for j = 1, 2, · · · , p, then N ← N + 1. Step 4. Repeat the second and third steps N times. Step 5. U2 (x) = N /N . 5 - Hybrid Intelligent Algorithm In order to compute the uncertain function U3 (x), we generate ωk from Ω according to the probability measure Pr and produce ξ(ωk ) for k = 1, 2, · · · , N .

N as N → ∞. Note that the sum k=1 h(x, ξ(ωk )) is just the number of ξ(ωk ) satisfying f (x, ξ(ωk )) ≥ f for k = 1, 2, · · · , N . Thus f is just the N th largest element in the sequence {f (x, ξ(ω1 )), f (x, ξ(ω2 )), · · · , f (x, ξ(ωN ))}, where N is the integer part of αN . 6 (Stochastic Simulation for U3 (x)) Step 1. Generate ω1 , ω2 , · · · , ωN from Ω according to the probability measure Pr. Equivalently, we generate random vectors ξ(ω1 ), ξ(ω2 ), · · · , ξ(ωN ) according to its probability distribution.

### Theory and Practice of Uncertain Programming by Boading Liu

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