Question $1$: $(40\backslash \%)$

A hiker plans to go on a camping trip. There are $5$ items the hiker wishes to take with her, but together they exceed the $\backslash (60-\backslash$mathrm$\{$lb$\}\backslash )$ weight limit she feels she can carry. To assist herself in the selection process, she has assigned a value to each item.

$($a$)$ Build a binary integer programming model and use a computer to solve it$.$

$($b$)$ Use the branch and bound method to solve this problem step by step. $($Hint: Each item is either chosen or not, you should add more constraints to the original BIP model, then solve this relaxed BIP model as the first linear programming model$).$ You should state clearly each subproblem $($a linear programming model$),$ its upper bound and lower bound, its optimal solution, and its further branches $($subproblems$),$ etc.

$($c$)$ Draw a diagram $($a tree$)$ to show all subproblems $($nodes$),$ branches, etc.