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knapsack problem

Can anyone explain how the writer rewrite the objective function in (7) to be written as (8)?

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    $\begingroup$ I deleted my "answer" (which was supposed to be a comment to Rob Pratt's answer). Please accept his answer rather than mine. $\endgroup$
    – prubin
    Feb 17 at 17:29

1 Answer 1

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Just consider the two cases:

  • If $z_i=0$, the LHS and RHS are both $0$.
  • If $z_i=1$, the LHS and RHS are both $p_i$.

Explicit calculations for the RHS: \begin{align} \theta_i \left(1 - \left(\frac{\theta_i-p_i}{\theta_i}\right)^0 \right) &= \theta_i \left(1 - 1\right) = 0 \\ \theta_i \left(1 - \left(\frac{\theta_i-p_i}{\theta_i}\right)^1 \right) &= \theta_i \left(1 - 1 + \frac{p_i}{\theta_i}\right) = \theta_i \cdot \frac{p_i}{\theta_i} = p_i \end{align}

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