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 \( t \)). \[ \text{max} \sum_{t=1}^{T} \sum_{i=1}^{m} (r_i S_{it} - h_i I_{it}) \] subject to \[ S_{it} \ge d_{it} \forall i, t \] <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