# dp1 - Deterministic Dynamic Programming Dynamic Programming...

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Deterministic Dynamic Programming Dynamic Programming (DP) determines the optimum solution to an n-variable problem by decomposing it into n stages with each stage constituting a single-variable sub problem. Recursive Nature of Computations in DP Computations in DP are done recursively, in the sense that the optimum solution of one sub problem is used as an input to the next sub problem.

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By the time the last sub problem is solved, the optimum solution for the entire problem is at hand. The manner in which the recursive computations are carried out depends on how we decompose the original problem. In particular, the sub problems are normally linked by common constraints. As we move from one sub problem to the next, the feasibility of these common constraints must be maintained.
We illustrate with the famous STAGECOACH problem. It concerns a mythical fortune seeker in Missouri who decided to go west to join the gold rush in California during the mid-19 th century. The journey would require travelling by stagecoach through different states. The possible choices are shown in the figure below. Each state is represented by a circled letter and the direction of

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travel is always from left to right in the diagram. Thus, four stages were required to travel from the point of embarkation in state A (Missouri) to his destination in state J (California). The distances between two states are also shown. Thus the problem is to find the shortest route the fortune-seeker should take.
A B E H J F C D G I 2 3 4 4 3 1 3 3 4 3 7 1 4 6 3 2 4 6 3 4

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A B E H J F C D G I 2 3 4 4 3 1 3 3 4 3 7 1 4 6 3 2 4 6 3 4 3 J 4 J 4 H 7 I 6 H 11 E or F 7 E 8 E or 11 C or D
Thus the optimum route will be A C D E F H I J i.e. A C E H J with optimum value 11. or A D E H J or A D F I J

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Formulation Let the decision variables y n (n=1,2,3,4) be the immediate destination on stage n. Thus the route selected is where y 4 =J A y 1 y 2 y 3 y 4 Now we do the same problem by Dynamic programming.
Let f n ( x n , y n ) be the total cost of the best overall policy for the remaining stages, given that the fortune seeker is in state x n , ready to start stage n, and selects y n as the immediate destination. Given n and x n , let y* n denote any value of y n (not necessarily unique) that minimizes f n ( x n , y n ) and let F n ( x n ) be the corresponding minimum value of     ( , ) n n n f x y

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Thus * ( ) min ( , ) ( , ) n n n n n n n n F x f x y f x y = = f n ( x n , y n ) = immediate cost (stage n) + minimum future cost (stages n+1 onward) , 1 1 ( ) n n x y n n c F x + + = + and x n +1 = T n ( x n , y n ) , state into which the system is transformed by the choice of y n . where
The values of for various x n and y n The objective is to find F 1 (A) and the corresponding route. DP finds it by successively finding F 4 ( x 4 ), F 3 ( x 3 ), F 2 ( x 2 ) for each of the possible states x i and then using F 2 ( x 2 ) to solve for F 1 (A).

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## This note was uploaded on 09/24/2010 for the course MATHEMATIC AAOC C222 taught by Professor Prof.dilipkumarsatpathi during the Fall '07 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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dp1 - Deterministic Dynamic Programming Dynamic Programming...

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