HW7_solution

HW7_solution - V n ( x,s ) = max y ≥ x { R · E [ miny,D...

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solution of homework 7 Problem 1 Let V n ( x,d ) denote the minimum cost from period n to the end of the plan- ning horizon, given x is the current capacity level before making capacity deci- sion at the beginning of period n, and d is the demand for this period. So we could have the optimality formulation as below: V n ( x,d ) = min y x { C n ( y - x ) + e n ( d - y ) + - e n ( y - d ) + + E [ V n +1 ( y, (1 + I n ) d )] } = min y x { C n ( y - x ) + e n ( d - y ) + E [ V n +1 ( y, (1 + I n ) d )] } where E [ V n +1 ( y, (1 + I n ) d )] = k =0 V n +1 ( y, (1 + k ) d ) × e - 0 . 1 × 0 . 1 k k ! . And the boundary condition is V N +1 ( x,d ) 0. We want to find V 1 (0 ,D 1 ) Problem 2 Let V n ( x,s ) denote the present value of cash flow from period n to the end of planning horizon, given that x is the current inventory level and s is the cash level. So we could have the optimality formulation as below:
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Unformatted text preview: V n ( x,s ) = max y ≥ x { R · E [ miny,D n ] + (1 + r 1 )( s-W ( y-x )) +-(1 + r 2 )( W ( y-x )-s ) +-W ( y-x ) + 1 1 + r 1 · V n +1 (( y-D n ) + ,Q ) } Where Q = s + r 1 ( s-W ( y-x )) +-r 2 ( W ( y-x )-s ) +-W ( y-x ) The boundary condition is: V N ( x,s ) = max y ≥ x { R · E [min { y,D N } ] + (1 + r 1 )( s-W ( y-x )) +-(1 + r 2 )( W ( y-x )-s ) + + S · E [( y-D N ) + ]-W ( y-x ) } Where S is the salvage value of the product. So we want to find V 1 (0 , 0) based on the above optimality formulation. 1...
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This note was uploaded on 03/16/2012 for the course IEOR 466 taught by Professor Richard during the Spring '12 term at Columbia.

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