Lecture 11

# Dynamic recursive relationship the above steps can be

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Dynamic Recursive Relationship The above steps can be summarized as f ( i , s ) = minimum { c sq + f ( i +1, q ) }, for i =1, 2, 3, 4 (1) for all ( s , q ) in network f (2, 7) = min{7+ f (3, 8), 1+ f (3, 9)} = min{7+1, 1+4}=5, d 2 (7)= city 9 f (2, 5) = min{7+ f (3, 8), 5+ f (3, 9)} = min{7+1, 5+4}=8, d 2 (5)= city 8 f (1, 2) = min{10+ f (2, 5), 12+ f (2, 6)} = min{10+8, 12+4}=16, d 1 (2)= city 6 f (1,3) = min{5+ f (2,5), 10+ f (2,6), 7+ f (2,7) } = min{5+8, 10+4, 7+5}=12, d 1 (3)= city 7 f (1,4) = min{ 15+ f (1,6), 13+ f (1,7) } = min{15+4, 13+5}=18, d 1 (4)= city 7 f (0,1) = min{2+ f (1,2), 5+ f (1,3), 1+ f (1,4) } = min{2+15, 5+12, 1+18}=17, d 0 (1)= city 2 or 3 Optimal route? 1-2-6-8-10 2+12+3+1=18 1-3-7-9-10 5+7+1+4=17

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13 A Machine Replacement Problem A machine can be used for up to 3 months Cost for a new machine: 1000 Maintenance costs for a machine in different month: m 1 = 60, m 2 = 80, m 3 = 120 Used machine can be traded-in for a value depending on the age: s 1 = 800, s 2 = 600, s 3 = 500 Question When to replace the machine? Consider the total cost in a year
14 The DP Model Stage: Month t , t =1, 2, …, 12 State: x =1, 2, 3 The age of the current machine Decision: Whether or not to replace a machine of age x with a new machine at the end of the month t Value function: f ( t , x ): the minimum cost from month t to month 12 given the current age of the machine is x

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15 The DP Model Final condition trade-in the machine at the end of month 12 f (12, 1) =1000− s 1 + m 1 . f (12, k ) =− s k + m k for k =2, 3 DP recursion: f ( t , x ) for x =1, 2, 3 f ( t , 1)=1000+ m 1 + min{− s 1 + f ( t +1, 1), f ( t +1, 2)} f ( t , 2)= m 2 +min{− s 2 + f ( t +1, 1), f ( t +1, 3)} f ( t , 3)= m 3 s 3 + f ( t +1, 1): must trade-in Optimal solution f (1, 1): suppose we buy a new machine in January
16 Summary Dynamic optimization principle: regardless of the route taken to enter a particular state in stage n , the decisions by which you can reach the destination, i.e. from stage n to the end, must constitute an optimal policy for leaving that state Based on this principle, we can formulate a dynamic recursive

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