Dynamical Aspects in Fuzzy Decision Making
The book focuses on the recent dynamical development in fuzzy decision making. Various kinds of dynamics regarding not only time but also structure of systems are discussed in theory and applications. First, fuzzy dynamic programming is reviewed from a vi
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Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw, Poland E-mail: [email protected] http://www.springer.de/cgi-binlsearch_book.pl ?series =2941
Further volumes of this series can be found at our homepage. Vol. 50. F. Crestani and O. Pasi (Eds.) Soft Computing in Infonnation Retrieval, 2000 ISBN 3-7908-1299-4 Vol. 51. 1. Fodor, B. De Baets and P. Perny (Eds.) Preferences and Decisions under Incomplete Knowledge, 2000 ISBN 3-7908-1303-6 Vol. 52. E.E. Kerre and M. Nachtegael (Eds.) Fuzzy Techniques in Image Processing, 2000 ISBN 3-7908-1304-4 Vol. 53. O. Bordogna and O. Pasi (Eds.) Recent Issues on Fuzzy Databases, 2000 ISBN 3-7908-1319-2 Vol. 54. P. Sincak and J. Vascak (Eds.) Quo Vadis Computational Intelligence?, 2000 ISBN 3-7908-1324-9 Vol. 55. J.N. Mordeson, D.S. Malik and S.-C. Cheng FuZD(UO,,,,,UN_II xo)
=
max
UO,···,UN-l
=
[/1>Co (uo) !\ ... !\ /1>CN-l (UN-I) !\ E/1>CN (XN)]
(15)
where the fuzzy goal is viewed to be a fuzzy event in X whose (nonfuzzy) probability is (cf. Zadeh, 1968) E/1>CN(XN)
=
L
p(XN
xNEX
I XN-I, UN-I) . /1>CN(XN)
(16)
due to Kacprzyk and Staniewski (1980): find an optimal sequence of decisions uo, ... , uN- I to maximize the expectation of the fuzzy decision's membership function, i.e. /1>D(U(j, ... , UN-I
max
uo,···,UN-l
I xo) =
E[/1>co (uo) !\ ... !\ /1>CN-l (uN-d !\ /1>CN (XN)]
(17)
and these formulations are clearly not equivalent. Bellman and Zadeh's approach Since in (15) /1>CN-l (UN-I) !\ E/1>CN[J(XN-I, UN-I)] depend only on UN-I, the next two right-most terms depend only on UN-2, etc., the structure of (15) is essentially the same as that of (13), and the set of fuzzy dynamic programming recurrence equations is /1>CN-i(XN-i) = maxUN _i [/1>cN-i(UN-i)!\ E/1>CN-i+l(XN-i+I)] { E/1>CN-i+l(XN-i+I) = EXP(XN-i+l I XN-I,UN-d x N-1.+1 x /1>CN-i+l (XN-i+I); i = 1, ... , N
I:x .
(18)
and we consecutively obtain uN_i or, in fact, optimal policies aN_i such that UN - i = aN_i(xN-i),i = 1, ... , N.
8
Kacprzyk and Staniewski's approach To solve problem (17), we first introduce a sequence offunctions hi: X x X}=lU ------> [0,1] and gj: X x x{::;U------> [0,1]; i = 0,1, ... , N; j = 1, ... , N - 1; such that
hN(XN, uo,···, uN-d = /-LcD (uo) /\ ... /\ /-Lc N - 1 (UN-d/\ /\/-LD(UO, ... ,UN-li xo) gk(Xk, Uo,· .. ,Uk) = L:~=l hk+l (Si' Uo,· .. ,Uk) P(Si I Xk, Uk) hk(Xk, Uo,···, uk-d = maxUk gk(Xk, Uo,···, Uk) ho(xo)
(19)
= max go(Xo, uo) uD
If the consecutive decisions and states are Un, ... ,Uj and Xo, . .. ,Xj, respectively, then gj is the expected value of /-LD(. I xo) provided that the next decisions are optimal, i.e., uj+l'··· ,u N_l . It can be shown (Kacprzyk and Staniewski, 1980; Kacprzyk, 1983b, 1997) that there exist functions Wk : Xx Xf=l U ------> U such that hk(Xk, Uo, ... , uk-d = gk(Xk, Uo,···, Uk-l, Wk(Xk, Uo, ... , Uk-l)). Then, an optimal policy sought, a;, t = 0,1, ... , N - 1, is given by
ao = wo(xo)
l~~~~~'.~~~.~. ~~~~~:~~.(~.o?!.....
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