Lecture 11.ppt

Lecture 11 - Dynamic Programming 1 Investment Problem Decision variable Xij =1 means we allocate j units of investment to opportunity i y = unused

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1 Dynamic Programming
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2 Investment Problem Decision variable X ij =1 means we allocate j units of investment to opportunity i y = unused investment 0 }, 1 , 0 { 8 ,..., 1 for , 1 10 subject to 5 . 0 max 4 0 8 1 4 0 8 1 4 0 = = = + + ∑∑ ∑∑ = = = = = y X i X y jX y X r ij j ij i j ij i j ij ij
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3 The DP Method Solve the problem sequentially At each stage, one investment opportunity is considered Let f ( i , s ) be the maximum total profit from stage i , i +1,…, to stage 8, given s Dynamic Programming recursion f ( i , s ) = max d { r id + f ( i +1, s d ) | 0 ≤ d ≤ 4, d s } Initial condition f ( 8, s ) =max d { r 8d +0.5( s d ) | 0 ≤ d ≤ 4, d s } The optimal solution of the problem is obtained from f (1,10) : the maximum profit from stage 1 to stage 8 with 10 units of money initially available
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4 Dynamic Programming Formulation Problem includes a series of decisions Over “real” time, or simply a decision sequence Stage, state, decision, and value function Sometimes naturally, sometimes not obviously A recursive equation To link different stages An initial condition to make calculation of DP feasible
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Advantages of DP DP formulation is often the best way to gain understanding of a problem and its solution Computational efficiency The number of states we have to consider: 8*11, or I*S , where I is the number of stages and S is the number of possible states For each state, we need to compare up to 5 possible actions, or D decisions, in general Overall calculation: I * S * D More information can be obtained What happens if we only have 7 units of money? We only need to see the value of
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This note was uploaded on 01/05/2011 for the course IELM IELM202 taught by Professor D during the Fall '10 term at HKUST.

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Lecture 11 - Dynamic Programming 1 Investment Problem Decision variable Xij =1 means we allocate j units of investment to opportunity i y = unused

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