A factory produces $m (i= 1,2,\cdots, m)$ products, each of which requires processing on $n(j = 1,2,\cdots, n)$ 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 $T (t = 1, 2, \cdots, T)$ 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 $\text{t}$), $X_{it}$ (amount of product $i$ manufactured in time period $\text{t}$) and $I_{it}$ (amount of product $i$ held in inventory at the end of time period $\text{t}$).
$$\max \sum \limits_{t=1}^{t} \sum\limits_{i=1}^{m}\left ( r_{i}S_{it} - h_{i}I_{it} \right )$$
$$\text{subject to}$$
$$S_{it}\geq d_{it} \forall \:i,t$$
$$\text{<capacity constraint>}$$
$$\text{<inventory balance constraint>}$$
$$X_{it}, S_{it}, I_{it}\geq 0; I_{10}=0$$
The capacity constraints and inventory balance constraints for this formulation are
- $\sum \limits_{i}^{m}a_{ij}X_{it}\leq c_{jt} \: \forall \:j,t$ and $I_{it}=I_{i,t-1}+X_{it}-S_{it}\:\forall \:i,t$
- $\sum \limits_{i}^{m}a_{ij}X_{it}\leq c_{jt} \: \forall \:i,t$ and $I_{it}=I_{i,t-1}+X_{it}-d_{it}\:\forall \:i,t$
- $\sum \limits_{i}^{m}a_{ij}X_{it}\leq d_{it} \: \forall \:i,t$ and $I_{it}=I_{i,t-1}+X_{it}-S_{it}\:\forall \:i,t$
- $\sum \limits_{i}^{m}a_{ij}X_{it}\leq d_{it} \: \forall \:i,t$ and $I_{it}=I_{i,t-1}+S_{it}-X_{it}\:\forall \:i,t$