$$ solve this optimation problem $by$ hand

$3.(20)$ A special court commission appointed $to$ resolve a bitter fight over legislative redistricting has

proposed $6$ combinations $of$ the $5$ disputed counties that could form new districts. The following table

marks the counties composing each proposed district and shows the district's deviation from the equal$-$

population norm.

The court wants $to$ select a minimum total deviation collection $of$ districts that includes each county

exactly once. Formulate this problem $as$ a set partitioning ILP.

$4.(20)$ Consider the pure integer linear program. After solving the $LP$ relaxation, $we$ obtained the

optimal simplex dictionary $as$ follows:

$x_k\frac{?}{b}=ar(b{)}_k-{\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}}{\overset{?}{}}}{}_{kj}{x}_j$ for all kinB

$x_{j}0$ for all $j,$

where $B$ denotes the set $of$ basic variables and $N$ denotes the set $of$ nonbasic variables.

Suppose that $\frac{?}{\text{b}\text{a}\text{r}}(b{)}_{k}is$ fractional for some kinB. Let $f_{k0}$ denote the fractional part $of\frac{?}{\text{b}\text{a}\text{r}}(b{)}_k,$ and $f_{kj}$

denote the fractional part $of{}_{kj}.$ Then the classic Gomory fractional cut can $be$ obtained $as$

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}}{\overset{?}{}}}{f}_{kj}{x}_j{f}_{k0}$

which can $be$ strengthened $to$

${\displaystyle \underset{f_{kj}{f}_{k0}}{\overset{?}{}}}{f}_{kj}{x}_j+{\displaystyle \underset{f_{kj}>f_{k0}}{\overset{?}{}}}\frac{f_{k0}}{1-f_{k0}}(1-f_{kj})x_j{f}_{k0}$

$(a)(5)$ Show that the strengthened form $is$ tighter than the classic Gomory fractional cut.

$(b)(15)$ Explain how the strengthened form $of$ Gomory cut can $be$ derived.