GATE ME 2021 Set 2 — Question 39

A factory produces $m (i = 1, 2, ..., m)$ products, each of which requires processing on $n (j = 1, 2, ..., n)$ workstations. Let $a_{ij}$ be the amount of processing time that one unit of the $i^{th}$ product requires on the $j^{th}$ workstation. Let the revenue from selling one unit of the $i^{th}$ product be $r_i$ and $h_i$ be the holding cost per unit per time period for the $i^{th}$ product. The planning horizon consists of $T (t = 1, 2, ..., T)$ time periods. The minimum demand that must be satisfied in time period $t$ is $d_{it}$, and the capacity of the $j^{th}$ workstation in time period $t$ is $c_{jt}$. Consider the aggregate planning formulation below, with decision variables $S_{it}$ (amount of product $i$ sold in time period $t$), $X_{it}$ (amount of product $i$ manufactured in time period $t$) and $I_{it}$ (amount of product $i$ held in inventory at the end of time period $t$).$$\max \sum_{t=1}^{T} \sum_{i=1}^{m} (r_i S_{it} - h_i I_{it})$$subject to
$S_{it} \ge d_{it} \quad \forall \ i, t$
< capacity constraint >
< inventory balance constraint >
$X_{it}, S_{it}, I_{it} \ge 0; I_{i0} = 0$
The capacity constraints and inventory balance constraints for this formulation are