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# Use the branch-and-bound method to find the optimal solution to the following IP:\begin{aligned} \max z &=7 x_{1}+3 x_{2} \\ \text { s.t. } & 2 x_{1}+x_{2} \leq 9 \\ & 3 x_{1}+2 x_{2} \leq 13 \\ & x_{1}, x_{2} \geq 0 ; x_{1}, x_{2} \text { integer } \end{aligned}

Solution

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In order to solve this IP using the branch-and-bound method, we must first solve its LP relaxation. If all solutions of the LP relaxation are integers, then this solution is also the solution of the given IP. Since the solution of the LP relaxation is at some vertex of its feasible region, by computing the value of $z$ for these points, we conclude that $(x_{1},x_{2})=(13/3,0)$ is its optimal solution since it maximizes the value of $z$.

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