$($Inventory Management $-$ Part $2)$ Consider the following mathematical programming model for a singleproduct dynamic lot$-$sizing problem of $N$ periods.

Minimize $TRC={\displaystyle \underset{t=1}{\overset{N}{}}}(A{y}_t+I_{t}vr),$

Subject $to{I}_t=I_{t-1}+Q_t-D_t$ for

$,Q_{t}M{y}_t$ for $t=1,$dots,$N$

$,Q_{t}0,I_{t}0,y_{t}in\{0,1\}$ for $t=1,$dots,$N$

where $M$ is sufficiently large positive number

$($a$)$ Based on the following changes on the existing parameters, additional parameters and restrictions, reconstruct $($modify$)$ the above model.

The ordering $($setup$)$ cost in the periods may vary from one period to another.

The unit purchasing cost in the periods may vary from one period to another.

Backordering is allowed in all periods, except the last period.

Backordering cost per unit per period is $.$

The backorder quantity in a period can be at most $10$ percent of the cumulative demand for this period.

Maximum number of units that can be ordered $($lot size$)$ in each period is $Q_{\text{m}\text{a}\text{x}}.$

$($b$)$ Construct a $5-$period example problem. Using the reconstructed model in part $($a$),$ solve your $5-$period problem by GAMS.