$3$a$)$ Eight items with weights $8,3,5,5,9,6,7,$ and $12$ lbs respectively must be packed in three boxes. Each box has a capacity of $20$ lbs$.$ It would be desirable to distribute the load evenly between the three boxes.

$($i$)$ Formulate an Integer Linear Programme that can be used to solve this problem. Clearly define decision variables, objective function and constraints. $[30\%]$

$($ii$)$ Suppose items three and four cannot be packed in the same box. How would you modify your model? $[10\%]$

$($iii$)$ Suppose items one and two must be packed in the same box. How would you then modify your model? $[10\%]$