# sum of maxima vs the maximum of the sum

Consider the following integer program \begin{align} \max &\sum\nolimits_{i,j} U_i(j)\cdot x_{i,j}\\ \text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant c_j,& &\forall\, j\\ & x_{i,j}\in\{0,1\}. \end{align} and the following set $\Gamma$ of integer programs ($i$ is fixed below): \begin{align} \max &\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\ \text{subject to}& x_{i,j}\cdot f\left(i,j\right)\leqslant c_j,& &\forall\, j\\ & x_{i,j}\in\{0,1\}. \end{align} and this is done $\forall \,i$. I want to have the same results from both optimization problems(i.e. same $x_{i,j}$ from the first optimization problem and the second set and $\sum\nolimits_{i}\max\sum\nolimits_{j} U_i(j)\cdot x_{i,j}=\max\sum\nolimits_{i,j} U_i(j)\cdot x_{i,j})$

The problem is that the constraints are coupled as we can see. Consequently, the order in which the problems in $\Gamma$ are solved changes the answer. For example if we start with $i=1$ and do the maximization, we will find $x_{i,j}\, \forall j$. We can then set $c_j$ to $c_j-x_{i,j}\cdot f(i,j)$ and solve for the next $i$. But the order in which we choose the $i$'s is important. Intuitively, I am guessing that there should be an optimal order that leads to the same result. But, how can I find the optimal order? Does this problem has a well known name?

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Any ideas to help?? – john Jul 6 '12 at 19:32