Question $1.(25$ points$)$: $(Transportation$ problem$)$ A company supplies goods $to$ three customers, who

require $40,50$ and $40$ units respectively. The company has three warehouses, each $of$ which has $30$ units

available. The costs $of$ shipping $1$ unit from each warehouse $to$ each customer are shown $in$ the table below.

There $is$ a penalty for unmet demand: With customer $1,$ a penalty cost $of$ $$70$ per unit $is$ incurred; with

customer $2,$$$75$ per unit; and with customer $3,$$$65x_{ij}$ denotes the amount $to$

ship from warehouse $ito$ customer $j,$ and $c_{ij}is$ the corresponding unit shipping cost.min,${\displaystyle \underset{i=1}{\overset{3}{}}}{\displaystyle \underset{j=1}{\overset{3}{}}}{c}_{ij}{x}_{ij}+70(40-{\displaystyle \underset{i=1}{\overset{3}{}}}{x}_{i1})+75(50-{\displaystyle \underset{i=1}{\overset{3}{}}}{x}_{i2})+65(40-{\displaystyle \underset{i=1}{\overset{3}{}}}{x}_{i3})$

$s.t.{\displaystyle \underset{j=1}{\overset{3}{}}}{x}_{ij}30,i=1,$cdots,$3(supply$ constraints$)$

$,x_{ij}0,i=1,$cdots,$3,j=1,$cdots,$3$

$(b)$ Formulate the problem $as$ a balanced transportation problem.

$(c)$ Solve this problem $by$ either Python. Print out your code and the results you run $to$ attach with your

solution. For Python, you can modify the code that the instructor provided $to$ solve this problem.

You need $to$ provide your code, a screen shot $of$ the results when you run the code, and explain the meaning

$of$ solutions you obtain. Convert everything into $PDF$ and submit $it$ together with your $HW$ solution $ina$

single file.