Consider the multi$-$echelon inventory system described in the figure below. The planning

horizon consists of $5$ periods. The number of units of item needed to carry out a task are

colored in blue. The number of units of items produced by a task are colored in red. $($For

example, $2$ units of item $1$ and $8$ units of item $2$ are needed to carry out task $1,$ and $3$ units of

item $3,1$ unit of item $4$ and $7$ units of item $5$ are produced.$)$

Items $1$ and $2$ are raw materials, whose purchase costs are $2,$ and $3$ respectively in each period.

The variable cost of carrying out task $i$ in each period is $c_i$ and the fixed cost is $A_i,$ where

$(c_1,c_2,c_3)=(10,20,15)$ and $(A_1,A_2,A_3)=(100,400,300).$ The cost of carrying one unit of

item $r$ in inventory from one period to the next is given by $h_r,$ with $(h_1,h_2,h_3,h_4,h_5,h_6)=$

$(1,2,2,4,1,3).$ The demand for item $r$ in period $t,D_{rt}$ is specified in the table below:

The system manager decides on the quantity of each item to produce in each period to

minimize the production, inventory, purchase and setup costs, while meeting all the demand.

Assume no backorders are allowed.

$($a$)$ Formulate the lot sizing problem as a mixed integer linear program $($under the big

time$-$buckets assumption$).$

$($b$)$ Solve the MILP in part $($a$),$ and find the optimal solution. You may use any solver of

your choice; we recommend Julia$/$JuMP$.$