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GATE Mechanical Engineering (ME) 2021 Shift 2 Solved Paper

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Question : 39 of 65

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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).

\max \sum\limits_{t=1}^{T} \sum\limits_{i=1}^{m} (r_{i}S_{it} - h_{i}I_{it})

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

$\sum\limits_{i}^{m} a_{ij} X_{it} \leq c_{jt} \forall i,t$ 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 $I_{it}=I_{i,t-1}+X_{it}-S_{it} \forall i,t$
$\sum^{m} a_{ij} X_{it} \leq d_{it} \forall i,t$ and $I_{it}=I_{i,t-1}+S_{it}-X_{it} \forall i,t$
$\sum\limits_{i}^{m} a_{ij} X_{it} \leq c_{jt} \forall j,t$ and $I_{it}=I_{i,t-1}+X_{it}-S_{it} \forall i,t$

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