A factory produces m (i = 1, 2, ..., m) products, each of which requires processing on n (j = 1, 2, ..., n) workstations. Let aij be the amount of processing time that one unit of the ith product requires on the jth workstation. Let the revenue from selling one unit of the ith product be ri and hi be the holding cost per unit per time period for the ith 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 dit, and the capacity of the jth workstation in time period t is cjt. Consider the aggregate planning formulation below, with decision variables Sit (amount of product i sold in time period t), Xit (amount of product i manufactured in time period t) and Iit (amount of product i held in inventory at the end of time period t).\({\rm{max}}\mathop \sum \limits_{t = 1}^T \mathop \sum \limits_{i = 1}^m \left( {{r_i}{S_{it}} - {h_i}{I_{it}}} \right)\)Subject toSit ≥ dit ∀ i, t< capacity constraint >< inventory balance constraint >Xit, Sit, Iit ≥ 0; Ii0 = 0The capacity constraints and inventory balance constraints for this formulation respectively are

• A. \(\displaystyle\sum_i^m a_{ij}X_{it} \le c_{jt} \ \forall \ i, t\) and \(I_{it} = I_{i, t-1} + X_{it} - d_{it} \ \forall \ i, t\)
• B. \(\displaystyle\sum_i^m a_{ij}X_{it} \le d_{it} \ \forall \ i, t\) and \(I_{it} = I_{i, t-1} + X_{it} - S_{it} \ \forall \ i, t\)
• C. \(\displaystyle\sum_i^m a_{ij}X_{it} \le d_{it} \ \forall \ i, t\) and \(I_{it} = I_{i, t-1} + S_{it}- X_{it} \ \forall \ i, t\)
• D. \(\displaystyle\sum_i^m a_{ij}X_{it} \le c_{jt} \ \forall \ j, t\) and \(I_{it} = I_{i, t - 1} + X_{it} - S_{it} \ \forall \ i, t\)