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

Question

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 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 ∑ₜ₌₁^(T)∑_(i=1)^(m)(r_iS_(it)-h_iI_(it)) subject to S_it ≥ d_it ∀ i,t < capacity constraint > < inventory balance constraint > X_it, S_it, I_it ≥ 0; I_i0 = 0 The capacity constraints and inventory balance constraints for this formulation are

Accepted Answer

∑_(i=1)^(m) a_(ij)X_(it) ≤ c_(jt), ∀ j,t and I_(it)=I_(i,t-1)+X_(it)-S_(it), ∀ i,t. The correct option accurately represents the capacity constraints by summing the processing times of all products on each workstation to ensure they do not exceed the available capacity, while the inventory balance constraint correctly reflects the relationship between production, sales, and inventory levels over time.

Solution

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