Download Fuzzy Stochastic Multiobjective Programming by Masatoshi Sakawa PDF
By Masatoshi Sakawa
Although experiences on multiobjective mathematical programming lower than uncertainty were accrued and several other books on multiobjective mathematical programming below uncertainty were released (e.g., Stancu-Minasian (1984); Slowinski and Teghem (1990); Sakawa (1993); Lai and Hwang (1994); Sakawa (2000)), there appears to be like no e-book which matters either randomness of occasions with regards to environments and fuzziness of human judgments at the same time in multiobjective choice making difficulties. during this ebook, the authors are occupied with introducing the most recent advances within the box of multiobjective optimization lower than either fuzziness and randomness at the foundation of the authors’ carrying on with learn works. particular pressure is put on interactive choice making points of fuzzy stochastic multiobjective programming for human-centered structures below uncertainty in such a lot real looking events whilst facing either fuzziness and randomness. association of every bankruptcy is in brief summarized as follows:
Chapter 2 is dedicated to mathematical preliminaries, with a view to be used through the remainder
of the e-book. beginning with simple 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 selection maker’s (DM’s) judgment for stochastic
objective capabilities and/or constraints in multiobjective difficulties, fuzzy multiobjective stochastic
programming is built.
In bankruptcy four, in the course of 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 corporations the place there exist DMs who've various priorities in making judgements, two-level programming difficulties are mentioned.
Finally, bankruptcy 6 outlines a few destiny learn directions.
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Additional resources for Fuzzy Stochastic Multiobjective Programming
Sample text
N −1. 88) of the follower from j = 1 to j = N in turn. 88) is the Stackelberg solution. 88) is solved. 90) Ax + By ≤ b ⎪ ⎪ ⎭ y ≥ 0, uT ≥ 0, vT ≥ 0, where u is an m dimensional row vector and v is an n2 dimensional row vector. 91) ⎪ ⎪ Ax + By ≤ b ⎪ ⎪ ⎭ x ≥ 0, y ≥ 0, uT ≥ 0, vT ≥ 0. 91), v is eliminated and the equality constraint u(Ax + By − b) − vy = 0 is transformed into u(b − Ax − By) + (uB + d2 )y = 0. 92) implies that b − Ax − By ≥ 0, uT ≥ 0, (uB+d2 )T ≥ 0, y ≥ 0. Let Ai and Bi be the ith row vector of the matrix A and the matrix B, respectively, and let B j and d2 j be the jth column vector of the matrix B and the jth element of the vector d2 , respectively.
In this section, we consider a case where there are two DMs; one of the DMs first makes a decision, and then after acknowledging the decision of the first DM, the other DM chooses a decision. Such a situation is formulated as a two-level programming problem. A linear programming problem with two DMs is formulated as follows. For the sake of simplicity, we call the two DMs DM1 and DM2 in this subsection. Let x1 and x2 denote the column vectors of the decision variables of DM1 and DM2, respectively, and let z1 (x1 , x2 ) = c11 x1 + c12 x2 and z2 (x1 , x2 ) = c21 x1 + c22 x2 denote the objective functions of DM1 and DM2, respectively, where ci1 , i = 1, 2 are n1 dimensional coefficient row vector, and ci2 , i = 1, 2 are n2 dimensional coefficient row vector.
For two-level linear programming problems with cooperative DMs, fuzzy programming approaches have been developed (Lai, 1996; Shih, Lai and Lee, 1996; Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009). In fuzzy programming for two-level linear programming by Sakawa et al. 79) if z1i < zi (x) ≤ z0i μi (zi (x)) = 1 − z0 ⎪ z ⎪ i i ⎪ ⎩0 if zi (x) > z0i . 80) for DM2 ⎪ subject to Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. , n = n1 + n2 . Because the two DMs make decisions cooperatively, the decision variable vector is represented simply by x without partition.