Lab 9 - A[0,w] = 0 When there are no items to be...

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Lab 9 – Dynamic Programming 1. 0 1 Knapsack Problem The 0 1 Knapsack Problem is a common problem in Integer Programming. We are given a set of items, each with a certain value and weight. The problem is to choose the items to place into the knapsack (which has a certain weight limit) that maximized the total value. 2. IP Formulation The problem can be easily formulated into a Binary Integer Program: Let x i = 1 if we include item i in the knapsack and 0 otherwise Let v i and w i represent the value and weight of each item i Let W be the weight limit of the knapsack min ෍ ݔ ݒ ௜ୀଵ ݏݑܾ݆݁ܿݐ ݐ݋: ෍ ݔ ݓ ൑ܹ ௜ୀଵ 3. Dynamic Programming Formulation It is also possible to formulate the problem as a DP if the weights are integers: Let A[i,w] be the value of the best knapsack given we use up to items i and weight w The optimality equations and boundary conditions are as follows:
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Unformatted text preview: A[0,w] = 0 When there are no items to be considered, there is no value A[i, 0] = 0 When there is no weight remaining, there is no value A[i, w] = A[i 1,w ] ( if w i > w) New item being considered weighs more than the current limit A[i, w] = max(A[i 1,w], m[i 1,w w i ]+v i ) (if w i w) Calculating A[N,W] given there are N items and a total Weight of W gets the optimal solution to the DP 4. MATLAB Code A Dynamic Programming Problem has been written in MATLAB and can be downloaded from: http://www.mathworks.com/matlabcentral/fileexchange/22783 1 knapsack Download and look at the code to see the Dynamic Programming Algorithm at work. 5. Extension Study the MATLAB code for the Knapsack DP for next weeks lab....
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This note was uploaded on 04/20/2011 for the course MIE 376 taught by Professor Daniel during the Spring '11 term at University of Toronto- Toronto.

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