MHB Value of optimal combination of objects

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The discussion centers on the Knapsack problem, specifically how to determine the optimal combination of objects with given weights and values that fit into a sack of capacity W. The formula K(w) = max_{w_i ≤ w} {K(w - w_i) + v_i} is used to express the value of the optimal combination, where K(w) represents the maximum value achievable with weight w. Participants explore the reasoning behind this formula, illustrating how it builds on previous optimal combinations by considering the addition of each object. They also discuss how to derive values for specific weights and how to create a table to visualize the optimal combinations. The conversation concludes with a method for determining which objects to include for achieving the optimal combination.
  • #31
I like Serena said:
Suppose we had 2 identical objects with place for only 1 of them.
Then we would have 2 solutions.

Of suppose we had objects with weights 1 and 4, and also objects with weights 2 and 3, both adding up to the same value.
Then, if we only had a weight allowance of 5, we would still have 2 solutions. (Nerd)

I understand! Thank you very much! (Clapping)(Smirk)
 

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