New PDF release: Fuzzy Stochastic Multiobjective Programming

By Masatoshi Sakawa

ISBN-10: 1441984011

ISBN-13: 9781441984012

Although stories on multiobjective mathematical programming less than uncertainty were amassed and a number of other books on multiobjective mathematical programming lower than uncertainty were released (e.g., Stancu-Minasian (1984); Slowinski and Teghem (1990); Sakawa (1993); Lai and Hwang (1994); Sakawa (2000)), there appears no e-book which matters either randomness of occasions relating to environments and fuzziness of human judgments at the same time in multiobjective choice making difficulties. during this booklet, the authors are taken with introducing the newest advances within the box of multiobjective optimization below either fuzziness and randomness at the foundation of the authors’ carrying on with learn works. targeted pressure is put on interactive choice making elements of fuzzy stochastic multiobjective programming for human-centered platforms lower than uncertainty in so much reasonable occasions while facing either fuzziness and randomness. association of every bankruptcy is in short summarized as follows:

Chapter 2 is dedicated to mathematical preliminaries, so that it will be used through the remainder

of the ebook. beginning with uncomplicated notions and strategies of multiobjective programming, interactive

fuzzy multiobjective programming in addition to fuzzy multiobjective programming is outlined.

In bankruptcy three, via contemplating the imprecision of choice maker’s (DM’s) judgment for stochastic

objective services and/or constraints in multiobjective difficulties, fuzzy multiobjective stochastic

programming is constructed.

In bankruptcy four, throughout the attention of not just the randomness of parameters concerned in

objective services and/or constraints but in addition the specialists’ ambiguous figuring out of the discovered values of the random parameters, multiobjective programming issues of fuzzy random variables are formulated.

In bankruptcy five, for resolving clash of determination making difficulties in hierarchical managerial or

public firms the place there exist DMs who've diverse priorities in making judgements, two-level programming difficulties are mentioned.

Finally, bankruptcy 6 outlines a few destiny learn directions.

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Additional info for Fuzzy Stochastic Multiobjective Programming

Example text

Zˆk )T which reflects the desired values of the objective functions, and it is thought that by changing the reference points in the interactive solution procedure, the DM can perceive, understand and learn the DM’s own preference. After the reference point zˆ is specified, the following minimax problem is solved: minimize max {zi (x) − zˆi } 1≤i≤k subject to Ax ≤ b, x ≥ 0. 62) is a Pareto optimal solution closest to the reference point in the L∞ norm; the L∞ norm is also called the Tchebyshev norm or the Manhattan distance.

Let q+ and q− denote the penalty costs for making these errors, and let c be the original costs for the activities in the production plant. Then, the objective function to be minimized may be the expectation of cx + q+ y+ + q− y− . 43) Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0, where E means the function of expectation. 43) is called a simple recourse problem. 43) unless qi + q− i is negative, and therefore we assume − + q ≥ 0, i = 1, . . 44) ⎪ ⎪ ∑ ti j x j , ⎪ ⎭ j=1 where ti j is the i j-element of T . Assume that the random variables d¯i , i = 1, .

28) is simply denoted by Aiα = aLiα , aRiα . 26) can be represented as x ∈ Xpos (α) = x ≥ 0 | aLiα x ≤ bRiα , i = 1, . . , m . 23) are represented by L-R fuzzy numbers: C˜ j = (c j , β j , γ j )LR , j = 1, . . , n, A˜ i j = (ai j , δi j , εi j )LR , i = 1, . . , m, j = 1, . . , n, B˜i = (bi , ζi , ηi )LR , i = 1, . . , m; and L(y) = R(y) = max{0, 1 − |y|}. 30) can be rewritten by ordinary linear inequalities x ∈ Xpos (α) = x ≥ 0 n ∑ {ai j − (1 − α)δi j }x j ≤ bi + (1 − α)ηi , i = 1, . .

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