Dynamic Planning And Two Or Multistage Programming Under Risk
Dynamic Planning And Two Or Multistage Programming Under Risk
Michael Werner
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-8- Appendix EXAMPLE 1 ; Given M y = f ; f := vector of constants y > with the Moore-Penrose (Generalized)-Inverse M of the matrix M, the set of feasible solutions can be represented by (yj = M*f + [e - M* m] w > 0; w I 0. If the random variable nature has a continuous density function, we obtain for our example the optimization problem opt " 'i^^i (^1^^ +_/ g(P • S2y^(y^) d^} g(^) means the density functions of the random variable ? := b - A y . The set Tj of feasible solutions y is defined by... T^ = T^n (m^ y^ = b^ - A^ y-*-; y^ > } By using the Moore-Penrose-Inverse we obtain {y2] = m2* [b^ - a2 y'j +[e-m2*m2J„>0 ^ M^* [b^ - A^ y-*"] + [e - M^* M^J w >, the additional restrictions for y to reduce T to T~. If for example 2* 2 M is the left inverse of M they will be reduced to ^2*, 2 > ^2*, 2 1 M b i M Ay T~ is defined by T- = jy' I m' y^ = bl; m2* a2 yl i m2* b^; y^ t o] . EXAMPLE 2 : For our example with a discrete random variable nature the optinization problem can be written as -9- K y^ ^h ' ^21 y^^ + ••• + Pq • S2Q y^"^ -♦^p'^' 21 A^^ y' + M^^ y^l = b" A^^'y^ t '.
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