Q$2)$ While implementing the Branch & Bound algorithm, use Pyomo to solve the LP relaxation subproblems at each node $($all variable domains set to 'NonNegativeReals$).$

Among variables with fractional values, select the one with the smaller index for branching. Report the LB and UB at each node. Clearly state how the nodes are fathomed.

$($Note that the problem contains both integer and continuous variables.$)(50$ points$)$

Use the MIP branch$-$and$-$bound algorithm presented in

Sec. $12\mathrm{.}7$ to solve the following MIP problem interactively:

Maximize

Z $=5$x$,+4$x$2+4$x$3+2$x$4,$

subject to

x$,+3$x$2$

$5$x$+$

X$2$

$$

$2$

$+2$x$3$

$+$

$4$

$+2$x$4$

$53$

$+$

IA IA

$=$

$10$

$15$

$6$

and

$0,$

x$,$ is integer,

for $1,2,3,4$

for j $=1,2,3.$