$\backslash [$

x$\_\{$j$\}\backslash$text $\{$ is binary for $\}$ j$=1,2,\backslash$ldots$,5\backslash$text $\{.\}$

$\backslash ]$

$($Note: First use AMPL and CPLEX to solve this problem, making sure to specify the decision variables as binary. After obtaining the optimal binary $($integer$)$ solution this way, replace the binary restriction on the variables with $\backslash (>=0\backslash )$ and constraints, and use a branch$-$and$-$bound strategy to resolve the problem interactively. After solving the initial LP relaxation, you need to impose additional constraints as you branch and fill in the branch$-$and$-$bound tree. For each node in the tree, indicate the optimal solution of the corresponding LP relaxation at that node and its optimal value. Also, indicate where you fathom the tree and why. In addition to submitting the AMPL model, data, and run files, submit the final branch$-$and$-$bound tree; you do not have to submit the branch$-$and$-$bound tree computed during each step of the branch$-$and$-$bound algorithm.$)$