Lecture 11

# We only need to see the value of f 17 6 when dynamic

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We only need to see the value of f (1,7).

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6 When Dynamic Programming Fails When the outcome of a decision also depends on the correlation between different options, DP does not work Only knowing that there are 3 units of fund available for opportunity 8 is not enough to make the optimal decision for opportunity 8 When the outcome of any decision for 8 depends the decision made for opportunity 7
7 Best Route to Lhasa 10 9 8 7 6 5 2 3 4 1 Hong Kong Lhasa 1 5 2 13 15 10 7 12 5 10 3 7 4 5 7 1 1 4 You are in 1 2 3 4 stage 0

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8 Starting Somewhere in the Middle If you have reached city 6, which way should you take to reach Lhasa? 10 9 8 6 Lhasa 3 4 1 4 If City 6 is now your starting point? City 1 was your starting point Two stages left to reach 10 Minimum cost when in city 6 at stage 2: f (2, 6) From any city s at stage i , the minimum cost to reach the destination is f(i, s )
9 Why Dynamic Optimization? From any stage, the best route to reach the destination depends on which city you are currently in Dynamic optimization model tries to identify the rule (policy) by which you know what is the optimal decision, given your current stage and state

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10 The Optimality Principle An optimal policy must have the property that regardless of the route taken to enter a particular state, the remaining decisions must constitute an optimal policy for leaving that state 10 9 8 6 Lhasa 15 10 12 3 4 1 4 10 9 8 7 Lhasa 13 7 7 1 1 4 f (2, 6) = min{3+ f (3, 8), 4+ f (3, 9)} =min{4, 8} f (2, 7) = min{7+ f (3, 8), 1+ f (3, 9)} =min{8, 5} d 2 (6) = best of 8 or 9 d 2 (7) = best of 8 or 9 What solution approach does the principle suggest?
11 Starting from the End At the destination, f (4,10)=0, d 4 (10)=stop From city 8, f (3,8)=1+ f (4, 10)=1, d 3 (8)= city 10 from city 9, f (3,9)=4+ f (4,10)= 4, d 3 (9)=city 10 From city 6, f (2,6)=min{3+ f (3,8), 4+ f (3,9)}= min{3+1,4+4}=4, d 2 (6)= 8 10 Lhasa 1 4 10 9 8 Lhasa 3 4 1 4 10 9 8 5 Lhasa 5 10 7 5 1 4 round 1 round 2 round 3

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