Question $1$: Consider the transportation problem with the following data $(3$ points$)$:

$$

City $1$

City $2$

City $3$

Supply capacity

Supplier $1$

$5$

$3$

$11$

$50$

Supplier $2$

$4$

$14$

$7$

$30$

Supplier $3$

$10$

$6$

$9$

$45$

Demand

$40$

$25$

$60$

$$

$$

$(1\mathrm{.}1)$ Use the minimal cost method to find a feasible basic solution to the above problem;

$(1\mathrm{.}2)$Use Vogel$$s penalty method to find a feasible basic solution to the problem and use the simplex method if necessary to find the optimal solution. $$

$(1\mathrm{.}3)$ Find the optimal solution to the new problem where the capacity of supply $1$ increases to $55,$ and the demand d$3$ increases to $30.$

$(1\mathrm{.}4)$ Find the allowable range of change to c$23$ under which the current optimal solution remains optimal.