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The two sets of constraints are not the same, but both sets provide a formulation for the problem. The relationship between the two is that if you take a solution to (2.1) and (2.2), compute $z_t=\sum_i z_{ti}$, and substitute those values for $z_t$ everywhere in (1.1) and (1.2), keeping $\delta_t$ from the solution to (2.1) and (2.2), the constraints (1.1) ...

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