Test Academy | 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 hi 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 dit, and the capacity of the j th workstation in time period t is c jt . Consider the aggregate planning formulation below, with decision variables Sit (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). m a x ∑ t = 1 T ⁡ ∑ i = 1 m ⁡ ( r i S i t − h i I i t ) 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 respectively are ∑ i m a i j X i t ≤ c j t ∀ i , t and I i t = I i , t − 1 + X i t − d i t ∀ i , t , ∑ i m a i j X i t ≤ d i t ∀ i , t and I i t = I i , t − 1 + X i t − S i t ∀ i , t , ∑ i m a i j X i t ≤ d i t ∀ i , t and I i t = I i , t − 1 + S i t − X i t ∀ i , t , ∑ i m a i j X i t ≤ c j t ∀ j , t and I i t = I i , t − 1 + X i t − S i t ∀ i , t

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 rⁱ and hi 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 dit, and the capacity of the j^th workstation in time period t is c_jt. Consider the aggregate planning formulation below, with decision variables Sit (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).

$m a x \sum_{t = 1}^{T} \sum_{i = 1}^{m} (r_{i} S_{i t} - h_{i} I_{i t})$

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 respectively are

Options :

$\sum_{i}^{m} a_{i j} X_{i t} \leq c_{j t} \forall i, t$ and $I_{i t} = I_{i, t - 1} + X_{i t} - d_{i t} \forall i, t$
$\sum_{i}^{m} a_{i j} X_{i t} \leq d_{i t} \forall i, t$ and $I_{i t} = I_{i, t - 1} + X_{i t} - S_{i t} \forall i, t$
$\sum_{i}^{m} a_{i j} X_{i t} \leq d_{i t} \forall i, t$ and $I_{i t} = I_{i, t - 1} + S_{i t} - X_{i t} \forall i, t$
$\sum_{i}^{m} a_{i j} X_{i t} \leq c_{j t} \forall j, t$ and $I_{i t} = I_{i, t - 1} + X_{i t} - S_{i t} \forall i, t$

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